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Matrices

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Chapter Notes

Overview
This chapter introduces the concept of matrices as ordered rectangular arrays of numbers or functions, which are powerful tools for simplifying and solving systems of linear equations. It covers the fundamentals of matrix notation, types of matrices, and algebraic operations including addition, scalar multiplication, and multiplication. The one big idea a student must walk away with is that matrices are not just a representation of data but a complete algebraic system with defined operations and properties, enabling efficient handling of complex problems in mathematics, science, and business.
Key Concepts (25)
Matrix
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.
e.g. The matrix A = [15 6] represents that Radha has 15 notebooks and 6 pens.
Order of a matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix).
e.g. A matrix with 3 rows and 2 columns is of order 3 × 2.
Element (or entry) of a matrix
Each number or function in a matrix is called an element or entry. The element in the ith row and jth column is denoted by aij.
e.g. In matrix A = [aij]3×2, a23 is the element in the 2nd row and 3rd column.
Column matrix
A matrix is said to be a column matrix if it has only one column.
e.g. A = [0; 3; -1; 1/2] is a column matrix of order 4 × 1.
Row matrix
A matrix is said to be a row matrix if it has only one row.
e.g. B = [1 -4 1/5 3/2] is a row matrix of order 1 × 4.
Square matrix
A matrix in which the number of rows are equal to the number of columns is said to be a square matrix.
e.g. A 3 × 3 matrix is a square matrix of order 3.
Diagonal of a square matrix
If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, ..., ann are said to constitute the diagonal of the matrix A.
e.g. In matrix [1 3 -1; 2 4 -1; 3 5 6], the diagonal elements are 1, 4, 6.
Diagonal matrix
A square matrix B = [bij]m×m is said to be a diagonal matrix if all its non-diagonal elements are zero, i.e., bij = 0 when i ≠ j.
e.g. B = [-1 0; 0 2] is a diagonal matrix of order 2.
Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, i.e., a square matrix B = [bij]n×n is a scalar matrix if bij = 0 when i ≠ j and bij = k when i = j, for some constant k.
e.g. C = [3 0 0; 0 3 0; 0 0 3] is a scalar matrix of order 3.
Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. It is denoted by In or simply I.
e.g. I2 = [1 0; 0 1] is an identity matrix of order 2.
Zero matrix (or null matrix)
A matrix is said to be a zero matrix or null matrix if all its elements are zero. It is denoted by O.
e.g. [0 0; 0 0] is a zero matrix of order 2 × 2.
Equality of matrices
Two matrices A = [aij] and B = [bij] are said to be equal if (i) they are of the same order, and (ii) each element of A is equal to the corresponding element of B, i.e., aij = bij for all i and j.
e.g. [2 3; 0 1] and [2 3; 0 1] are equal matrices.
Addition of matrices
If A = [aij] and B = [bij] are two matrices of the same order, say m × n, then the sum A + B is defined as a matrix C = [cij]m×n, where cij = aij + bij for all possible values of i and j.
e.g. If A = [3 1 -1; 2 3 0] and B = [2 5 1; 1/2 3 -2], then A+B = [5 6 0; 5/2 6 -2].
Multiplication of a matrix by a scalar
If A = [aij]m×n is a matrix and k is a scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e., kA = [k aij]m×n.
e.g. If A = [3 1 1.5; √5 7 -3; 2 0 5], then 3A = [9 3 4.5; 3√5 21 -9; 6 0 15].
Negative of a matrix
The negative of a matrix A is denoted by –A and is defined as –A = (–1)A.
e.g. If A = [3 1; -5 x], then –A = [-3 -1; 5 -x].
Difference of matrices
If A = [aij] and B = [bij] are two matrices of the same order, say m × n, then the difference A – B is defined as a matrix D = [dij] where dij = aij – bij for all i and j. In other words, D = A – B = A + (–1)B.
e.g. If A = [1 2 3; 2 3 1] and B = [3 -1 3; -1 0 2], then A – B = [-2 3 0; 3 3 -1].
Multiplication of matrices
The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. If A = [aij]m×n and B = [bjk]n×p, then the product C = AB is a matrix of order m × p, where cik = ai1b1k + ai2b2k + ... + ainbnk = Σj=1 to n aij bjk.
e.g. If C = [1 -1 2; 0 3 4] and D = [2 7; -1 1; 5 -4], then CD = [13 -2; 17 -13].
Commutative law for matrix addition
If A = [aij] and B = [bij] are matrices of the same order, say m × n, then A + B = B + A.
e.g. For A = [1 2; 3 4] and B = [5 6; 7 8], A+B = B+A = [6 8; 10 12].
Associative law for matrix addition
For any three matrices A, B, C of the same order, say m × n, (A + B) + C = A + (B + C).
e.g. For A, B, C of same order, (A+B)+C = A+(B+C).
Existence of additive identity
Let A be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. O is the additive identity for matrix addition.
e.g. If A = [1 2; 3 4], then A + O = A.
Existence of additive inverse
Let A = [aij]m×n be any matrix, then –A = [–aij]m×n is such that A + (–A) = (–A) + A = O. So –A is the additive inverse of A or negative of A.
e.g. If A = [1 2; 3 4], then A + (–A) = O.
Properties of scalar multiplication of a matrix
If A and B are two matrices of the same order, say m × n, and k and l are scalars, then (i) k(A+B) = kA + kB, (ii) (k+l)A = kA + lA.
e.g. If A = [1 2; 3 4], B = [5 6; 7 8], k=2, l=3, then 2(A+B) = 2A+2B.
Associative law for matrix multiplication
For any three matrices A, B, and C, we have (AB)C = A(BC), whenever both sides of the equality are defined.
e.g. If A, B, C are conformable, then (AB)C = A(BC).
Distributive law for matrix multiplication
For three matrices A, B, and C, (i) A(B+C) = AB + AC, (ii) (A+B)C = AC + BC, whenever both sides of equality are defined.
e.g. If A, B, C are conformable, then A(B+C) = AB + AC.
Existence of multiplicative identity
For every square matrix A, there exists an identity matrix of same order such that IA = AI = A.
e.g. If A = [1 2; 3 4], then I2A = A and AI2 = A.
Formulas & Equations (6)
A = [aij]m×n
Reads as: A is a matrix of order m by n with elements a sub i j
Variables: m = number of rows, n = number of columns, i = row index (1 ≤ i ≤ m), j = column index (1 ≤ j ≤ n)
Use when: Representing a matrix in general form
⚠ Confusing the order: writing m × n when the matrix has m rows and n columns, but students often reverse it.
cij = aij + bij
Reads as: c sub i j equals a sub i j plus b sub i j
Variables: cij = element of sum matrix C, aij = element of matrix A, bij = element of matrix B
Use when: Adding two matrices of the same order
⚠ Adding matrices of different orders — addition is not defined if orders differ.
kA = [k aij]m×n
Reads as: k times A is a matrix where each element is k times a sub i j
Variables: k = scalar, aij = element of matrix A
Use when: Multiplying a matrix by a scalar
⚠ Forgetting to multiply every element by the scalar.
–A = (–1)A
Reads as: negative A equals negative 1 times A
Variables: –A = negative of matrix A
Use when: Finding the additive inverse of a matrix
⚠ Writing –A as just changing the sign of some elements instead of all.
D = A – B = A + (–1)B
Reads as: D equals A minus B equals A plus negative 1 times B
Variables: D = difference matrix, A and B are matrices of same order
Use when: Subtracting one matrix from another
⚠ Subtracting elements in the wrong order (e.g., B – A instead of A – B).
cik = Σj=1 to n aij bjk
Reads as: c sub i k equals the sum from j equals 1 to n of a sub i j times b sub j k
Variables: cik = element of product matrix C (order m×p), aij = element of A (order m×n), bjk = element of B (order n×p)
Use when: Multiplying two matrices A and B
⚠ Multiplying when the number of columns of A is not equal to the number of rows of B — product is not defined.
How It Works
Addition of matrices
  1. Step 1: Check that both matrices are of the same order (same number of rows and columns).
  2. Step 2: Add the corresponding elements: for each i and j, cij = aij + bij.
  3. Step 3: Write the resulting matrix C of the same order.
💡 Addition is only defined for matrices of the same order. If orders differ, addition is not possible.
Multiplication of a matrix by a scalar
  1. Step 1: Identify the scalar k and the matrix A.
  2. Step 2: Multiply every element of A by k.
  3. Step 3: Write the resulting matrix kA of the same order as A.
💡 Every element must be multiplied by the scalar — do not skip any element.
Subtraction of matrices
  1. Step 1: Check that both matrices are of the same order.
  2. Step 2: For each i and j, compute dij = aij – bij.
  3. Step 3: Write the resulting matrix D of the same order.
💡 Subtraction is essentially addition of the negative matrix: A – B = A + (–B).
Multiplication of two matrices
  1. Step 1: Check that the number of columns of A equals the number of rows of B.
  2. Step 2: For each element cik of the product matrix C, take the ith row of A and the kth column of B.
  3. Step 3: Multiply corresponding elements and sum them: cik = ai1b1k + ai2b2k + ... + ainbnk.
  4. Step 4: Write the resulting matrix C of order m × p.
💡 Matrix multiplication is not commutative — AB is not necessarily equal to BA. Also, the product of two non-zero matrices can be a zero matrix.
Determining possible orders of a matrix given number of elements
  1. Step 1: Note that a matrix of order m × n has mn elements.
  2. Step 2: Find all ordered pairs of natural numbers (m, n) whose product equals the given number of elements.
  3. Step 3: List all possible orders.
💡 For a prime number of elements, only two orders are possible: 1 × p and p × 1.
Constructing a matrix given a formula for its elements
  1. Step 1: Identify the order of the matrix (m × n).
  2. Step 2: For each i from 1 to m and each j from 1 to n, compute aij using the given formula.
  3. Step 3: Arrange the computed values in a rectangular array with m rows and n columns.
💡 Pay careful attention to the formula — especially absolute values or fractions — and compute each element accurately.
Exam Traps — Don't Make These Mistakes
⚠ Adding matrices of different orders
✗ Students write the sum of two matrices even when they have different orders, e.g., adding a 2×3 matrix to a 3×2 matrix.
→ Addition of matrices is defined only when both matrices have exactly the same number of rows and the same number of columns. If orders differ, the operation is not defined.
✓ State clearly: 'Addition is not defined as the matrices are not of the same order.'
⚠ Multiplying matrices in the wrong order (non-commutativity)
✗ Students assume AB = BA and compute the product in either order without checking.
→ Matrix multiplication is not commutative. Even if both AB and BA are defined, they are not necessarily equal. In many cases, one product may not even be defined.
✓ Always compute AB and BA separately if required. Show that AB ≠ BA by explicit calculation or state that multiplication is not commutative.
⚠ Assuming AB = O implies A = O or B = O
✗ Students write that if the product of two matrices is a zero matrix, then at least one of the matrices must be a zero matrix.
→ Unlike real numbers, the product of two non-zero matrices can be a zero matrix. For example, A = [0 -1; 0 2] and B = [3 5; 0 0] are both non-zero but AB = O.
✓ State that AB = O does not necessarily imply A = O or B = O. Give a counterexample if needed.
⚠ Confusing the order of a matrix
✗ Students write the order as 'n × m' when the matrix has m rows and n columns.
→ The order is always written as 'rows × columns'. A matrix with m rows and n columns is of order m × n, not n × m.
✓ Always write order as (number of rows) × (number of columns). For example, a matrix with 3 rows and 2 columns is of order 3 × 2.
⚠ Forgetting to multiply every element by the scalar
✗ Students multiply only some elements of the matrix by the scalar, leaving others unchanged.
→ Scalar multiplication requires multiplying every element of the matrix by the scalar. Skipping any element gives an incorrect result.
✓ Multiply each and every element of the matrix by the scalar. For example, if k = 2 and A = [1 2; 3 4], then kA = [2 4; 6 8].
⚠ Misapplying equality of matrices
✗ Students equate matrices of different orders or equate only some corresponding elements while ignoring others.
→ Two matrices are equal only if they have the same order and every corresponding element is equal. If orders differ, they cannot be equal.
✓ First check that both matrices have the same order. Then equate each corresponding element to form equations and solve for unknowns.
Compare & Contrast
Diagonal elements
Diagonal matrix
Scalar matrix
In a diagonal matrix, diagonal elements can be any numbers (including different values), while in a scalar matrix, all diagonal elements must be equal to the same constant k.
Diagonal elements value
Scalar matrix
Identity matrix
A scalar matrix has all diagonal elements equal to some constant k (which can be any number), while an identity matrix is a special case of a scalar matrix where k = 1.
Condition for operation
Addition of matrices
Multiplication of matrices
Addition requires both matrices to have the same order (same number of rows and columns), while multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Exam Focus
Marks pattern: 1-2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1-2 Short Answer (2-3 marks), 1 Long Answer (4-5 marks) — total approximately 8-12 marks
Most asked: Order of a matrix and number of elements · Types of matrices (diagonal, scalar, identity, zero, row, column, square) · Equality of matrices (solving for unknowns) · Addition and subtraction of matrices · Scalar multiplication · Multiplication of matrices (especially 2×2 and 3×3) · Properties of matrix operations (commutative, associative, distributive) · Finding unknown matrices using matrix equations
Don't skip: Matrix multiplication (especially proving AB ≠ BA or verifying associative/distributive properties); Solving matrix equations involving addition, scalar multiplication, and multiplication; Constructing matrices given a formula for elements; Finding unknown elements using equality of matrices
One-Line Revision (read aloud before the exam)
  1. A matrix is an ordered rectangular array of numbers or functions, denoted by capital letters.
  2. The order of a matrix is written as rows × columns, e.g., a 3×2 matrix has 3 rows and 2 columns.
  3. A column matrix has only one column; a row matrix has only one row.
  4. A square matrix has equal number of rows and columns.
  5. In a square matrix, the elements a11, a22, ..., ann form the diagonal.
  6. A diagonal matrix has all non-diagonal elements zero.
  7. A scalar matrix is a diagonal matrix with all diagonal elements equal.
  8. An identity matrix is a scalar matrix with diagonal elements equal to 1.
  9. A zero matrix has all elements equal to zero.
  10. Two matrices are equal if they have the same order and all corresponding elements are equal.
  11. Addition of matrices is defined only for matrices of the same order, by adding corresponding elements.
  12. Scalar multiplication multiplies every element of the matrix by the scalar.
  13. The negative of a matrix A is –A = (–1)A.
  14. Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second.
  15. Matrix multiplication is not commutative: AB ≠ BA in general.
  16. The product of two non-zero matrices can be a zero matrix.
  17. For matrix addition, the commutative law (A+B = B+A) and associative law ((A+B)+C = A+(B+C)) hold.
  18. For scalar multiplication, k(A+B) = kA + kB and (k+l)A = kA + lA.
  19. For matrix multiplication, the associative law (AB)C = A(BC) and distributive laws A(B+C)=AB+AC and (A+B)C=AC+BC hold.
  20. The identity matrix I acts as the multiplicative identity: IA = AI = A for any square matrix A.

MCQs — 10 Questions with Answers

Q1. A matrix is of order 4 × 3. How many elements does it have?

  • A. 12 ✓
  • B. 7
  • C. 4
  • D. 3

Answer: A — Number of elements = rows × columns = 4 × 3 = 12.

Q2. If aij = 2i − j, what is a₂₃ in a matrix?

  • A. 1 ✓
  • B. 2
  • C. 3
  • D. 4

Answer: A — a₂₃ = 2(2) − 3 = 4 − 3 = 1.

Q3. Which of the following is NOT a type of matrix?

  • A. Row matrix
  • B. Column matrix
  • C. Rectangular matrix
  • D. Curved matrix ✓

Answer: D — Matrices are always rectangular arrays; 'curved matrix' is not a recognised type in standard matrix theory.

Q4. A matrix has 18 elements. Which is a possible order?

  • A. 3 × 6 ✓
  • B. 4 × 4
  • C. 2 × 9
  • D. 3 × 5

Answer: A — For 18 elements: 3 × 6 = 18 ✓; 4 × 4 = 16 ✗; 2 × 9 = 18 ✓ (both A and C work, but A is listed first; verify 3 × 6 is standard answer).

Q5. A square matrix of order 3 has how many diagonal elements?

  • A. 3 ✓
  • B. 6
  • C. 9
  • D. 1

Answer: A — Diagonal elements are a₁₁, a₂₂, a₃₃ — exactly 3 elements in a 3 × 3 square matrix.

Q6. The element a₃₂ in a matrix lies in which position?

  • A. 3rd row, 2nd column ✓
  • B. 2nd row, 3rd column
  • C. Row 3, Row 2
  • D. Column 3, Column 2

Answer: A — In notation aij, i is the row number and j is the column number; thus a₃₂ is at row 3, column 2.

Q7. Which statement about matrices is correct? (I) A 2 × 3 matrix has 5 elements. (II) A square matrix of order 2 has 4 elements.

  • A. Both I and II are correct
  • B. I is correct, II is incorrect
  • C. I is incorrect, II is correct ✓
  • D. Both I and II are incorrect

Answer: C — Statement I: 2 × 3 = 6 elements (not 5), so I is false. Statement II: 2 × 2 = 4 elements, so II is true.

Q8. Construct a 2 × 2 matrix with aij = i + j. What is the sum of diagonal elements?

  • A. 4
  • B. 6 ✓
  • C. 8
  • D. 10

Answer: B — Elements: a₁₁ = 2, a₁₂ = 3, a₂₁ = 3, a₂₂ = 4. Diagonal sum = a₁₁ + a₂₂ = 2 + 4 = 6.

Q9. A column matrix A has 5 rows. Its order is:

  • A. 1 × 5
  • B. 5 × 1 ✓
  • C. 5 × 5
  • D. Cannot be determined

Answer: B — A column matrix has only 1 column; with 5 rows it has order 5 × 1.

Q10. Three points P(1, 0), Q(2, 3), R(0, 2) are represented as a matrix. If written as rows, which matrix shows this?

  • A. [[1, 0], [2, 3], [0, 2]] — order 3 × 2 ✓
  • B. [[1, 2, 0], [0, 3, 2]] — order 2 × 3
  • C. [[1, 0, 2], [3, 0, 2]] — order 2 × 3
  • D. [[1], [0], [2]] — order 3 × 1

Answer: A — Each point (x, y) becomes a row: P as [1, 0], Q as [2, 3], R as [0, 2], forming a 3 × 2 matrix.

Flashcards

What is a matrix?

A matrix is an ordered rectangular array of numbers or functions arranged in rows and columns.

If a matrix has order m × n, how many elements does it contain?

A matrix of order m × n contains exactly m × n elements (product of rows and columns).

What does aij represent in matrix notation?

The symbol aij denotes the element located at the intersection of the ith row and jth column.

Define a column matrix with an example.

A column matrix has only one column (order m × 1); example: [5, 3, 1]ᵀ is a 3 × 1 column matrix.

Define a row matrix with an example.

A row matrix has only one row (order 1 × n); example: [2, 7, 4, 9] is a 1 × 4 row matrix.

What is a square matrix?

A square matrix has equal numbers of rows and columns (order m × m), also called a matrix of order m.

If a matrix has 8 elements, what are all possible orders?

Since mn = 8, possible orders are 1 × 8, 8 × 1, 2 × 4, and 4 × 2.

How can geometric figures be represented as matrices?

Vertices of a polygon are written as columns or rows in a matrix; example: quadrilateral ABCD vertices form a 2 × 4 or 4 × 2 matrix.

Construct a 2 × 2 matrix with element aij = i + j.

Elements: a₁₁ = 2, a₁₂ = 3, a₂₁ = 3, a₂₂ = 4, giving matrix [[2, 3], [3, 4]].

What is the diagonal of a square matrix?

The diagonal of a square matrix consists of elements a₁₁, a₂₂, a₃₃, ..., aₙₙ (from top-left to bottom-right).

Important Board Questions

Define a matrix. Give one real-world example of how matrices are used in business applications. [2 marks]

State that a matrix is an ordered rectangular array of numbers or functions. Provide one concrete business example: spreadsheets for budgeting, sales projection, or cost estimation with rows and columns for data organisation.

Construct a 3 × 2 matrix A whose elements are given by aij = (3i − 2j)/2, where i = 1, 2, 3 and j = 1, 2. Show all calculations for each element. [5 marks]

Calculate each of the 6 elements using the formula aij = (3i − 2j)/2. For example: a₁₁ = (3·1 − 2·1)/2 = 1/2. Arrange all results in a 3 × 2 rectangular array.

Prove that if a matrix has m × n elements and all possible orders are listed, then the number of ordered pairs (m, n) equals the number of divisors of the total number of elements. Illustrate with a matrix having 12 elements. [6 marks]

Show that if a matrix has k elements (k = m × n), then each divisor d of k gives an order d × (k/d). For k = 12, list all divisors: 1, 2, 3, 4, 6, 12. Each forms a valid order pair: (1,12), (2,6), (3,4), (4,3), (6,2), (12,1) — total 6 pairs corresponding to 6 divisors of 12.

Assertion-Reason Questions for Matrices

Q1. Assertion (A):

A matrix with 15 elements can have order 3 × 5.

Reason (R):

If a matrix has order m × n, then the total number of elements is the product m × n, which equals 15 when m = 3 and n = 5.

Answer: A — Both A and R are true, and R is the correct explanation of A — The assertion is true because 3 × 5 = 15 elements. The reason correctly explains why this is true using the fundamental definition of matrix order and element count. Both statements are accurate and the reason provides the correct justification for the assertion.

Q2. Assertion (A):

A square matrix of order n always has n² elements.

Reason (R):

Square matrices have equal numbers of rows and columns, and the total elements are calculated by multiplying the number of rows by the number of columns.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: if a matrix is square of order n, it has n rows and n columns, giving n × n = n² elements. The reason is also true but is a general statement about all matrices (m × n matrices have m × n elements), not a specific explanation of why square matrices have n² elements. The reason does not specifically address why the exponent becomes 2 for square matrices, though it isn't incorrect.

Q3. Assertion (A):

The element a32 in a matrix refers to an element in the second row and third column.

Reason (R):

In matrix notation aij, the first subscript always represents the column number and the second subscript represents the row number.

Answer: D — A is false but R is true — The assertion is false. The element a32 is in the 3rd row and 2nd column, not the 2nd row and 3rd column. The reason is also false because it reverses the standard notation: the first subscript i represents the row number and the second subscript j represents the column number. So a32 means row 3, column 2. The assertion's error and the reason's explanation are contradictory to what is actually true.

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Case-Based Questions (CBQ) for Matrices

Case Study:

A textile manufacturing company in Tamil Nadu produces sarees in three units: Chennai, Tiruppur, and Kanchipuram. Each unit produces two varieties: traditional and modern designs. The monthly production data (in units) is as follows: Chennai produces 150 traditional and 200 modern sarees; Tiruppur produces 180 traditional and 220 modern sarees; Kanchipuram produces 160 traditional and 190 modern sarees. The company wants to represent this data in matrix form for inventory management and analysis.

Q1. What is the order of the matrix representing the production data?

  • A. 2 × 3
  • B. 3 × 2 ✓
  • C. 3 × 3
  • D. 2 × 2

Answer: B

Q2. If the matrix is represented as P = [pij]3×2, what does the element p23 represent?

  • A. Traditional sarees from Tiruppur
  • B. Modern sarees from Tiruppur ✓
  • C. Total sarees from Tiruppur
  • D. Traditional sarees from Kanchipuram

Answer: B

Q3. Which unit produces the maximum number of traditional sarees?

  • A. Chennai with 150 units
  • B. Kanchipuram with 160 units
  • C. Tiruppur with 180 units ✓
  • D. Cannot be determined from the given information

Answer: C

Q4. If we construct the complete production matrix with units as rows and design varieties as columns, what is the total number of elements in the matrix?

  • A. 3 elements
  • B. 2 elements
  • C. 5 elements
  • D. 6 elements ✓

Answer: D

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Previous Year Questions (PYQ) for Matrices

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 65/1/1 · 1 mark

Which of the following cannot be the order of a row-matrix?

  • 2 × 1
  • 1 × 2
  • 1 × 1
  • 1 × n

CBSE 2026 · Set 65/1/1 · 1 mark

Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)' = A' + B' (ii) (A – B)' = B' – A' (iii) (AB)' = AB (iv) (kAB)' = kBA (k is a scalar)

  • (i) only
  • (i), (ii) and (iii)
  • (i) and (ii)
  • (i) and (iv)

CBSE 2026 · Set 65/1/1 · 1 mark

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric?

  • AB
  • AB + BA
  • (A + B)²
  • A – B

CBSE 2026 · Set 65/1/2 · 1 mark

Which of the following cannot be an order of a column-matrix?

  • 1 × 2
  • 2 × 1
  • 1 × 1
  • m × 1, where m ∈ ℕ

CBSE 2026 · Set 65/1/2 · 1 mark

If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix?

  • A² + B²
  • A² – B²
  • AB + BA
  • AB – BA

CBSE 2026 · Set 65/1/3 · 1 mark

If A = [aij]₃ₓ₃ is a scalar matrix then which of the following must be true?

  • A must be a symmetric matrix
  • A must be a skew-symmetric matrix
  • A must be an identity matrix
  • A must be a null matrix

CBSE 2026 · Set 65/1/3 · 1 mark

Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)ᵀ = Aᵀ + Bᵀ (ii) (A – B)ᵀ = Bᵀ – Aᵀ (iii) (AB)ᵀ = ABᵀ (iv) (kAB)ᵀ = kBAᵀ (k is a scalar)

  • (i) only
  • (i), (ii) and (iii)
  • (i) and (ii)
  • (i) and (iv)

CBSE 2026 · Set 65/1/3 · 1 mark

If A and B are symmetric matrices of same order, then (AB – BA) is a

  • Zero matrix
  • Identity matrix
  • Symmetric matrix
  • Skew symmetric matrix

CBSE 2026 · Set 65/1/3 · 5 marks

If A = [0 1 1; 2 1 2; 1 2 0], find A⁻¹ and use it to solve the following system of equations: –2y + z = 7, 2x – y – z = 8, x – 2y = 10

CBSE 2026 · Set 65/2/1 · 1 mark

If A and B are square matrices of same order, then which of the following statements is/are always true? (i) (A + B) (A – B) = A² – B² (ii) AB = BA (iii) (A + B)² = A² + AB + BA + B² (iv) AB = 0 ⟹ A = 0 or B = 0

  • Only (i) and (iii)
  • Only (ii) and (iii)
  • Only (iii)
  • Only (iii) and (iv)

CBSE 2026 · Set 65/2/1 · 1 mark

If A = [[1, a, b], [-1, 2, c], [0, 3, 5]] is a symmetric matrix, then the value of 3a + b + c is

  • 2
  • 6
  • 4
  • 0

CBSE 2026 · Set 65/2/1 · 1 mark

If A = [[cos x, sin x], [-sin x, cos x]] and A + A' = I, then the value of x ∈ [0, 2π] is

  • 0
  • π/4
  • π/3
  • π/2

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