Matrices — Class 12 MCQs with Answers

Matrices — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Order of matrix with 2 rows, 3 columns:

   A6B3 × 2C2 × 3D5
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   Answer: 2 × 3

   Explanation: Matrix order is always written rows × columns, so 2 rows and 3 columns gives 2×3 — not 3×2, which swaps the convention.

2. Q2 of 6Easy

   A row matrix has:

   ASame rows and columnsB1 columnC1 rowDNo restriction
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   Answer: 1 row

   Explanation: A row matrix has exactly 1 row and any number of columns — don't confuse it with a column matrix, which instead has exactly 1 column.

3. Q3 of 6Medium

   For AB to be defined, columns of A must equal:

   ARows of BBColumns of BCRows of ADDeterminant
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   Answer: Rows of B

   Explanation: For AB to exist, the inner dimensions must match: columns of A equal rows of B, since (m×n)(n×p) → m×p; the column count of B is irrelevant here.

4. Q4 of 6Medium

   Assertion (A): The 3 × 3 matrix A = [aᵢⱼ] with aᵢⱼ = i − j is skew symmetric. Reason (R): For any square matrix A, the matrix A + A′ is symmetric.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: aⱼᵢ = j − i = −(i − j) = −aᵢⱼ for all i, j, so A′ = −A. R is true by Theorem 1. But R is about the sum A + A′, not about why this particular A satisfies aⱼᵢ = −aᵢⱼ, so R does not explain A.

5. Q5 of 6Hard

   Any square matrix can be written as:

   AProduct of two diagonalBSum of two symmetricCSum of symmetric and skew-symmetric matricesDInverse of itself
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   Answer: Sum of symmetric and skew-symmetric matrices

   Explanation: Any square matrix splits as A = (A+Aᵀ)/2 + (A−Aᵀ)/2, where the first term is symmetric and the second is skew-symmetric — a genuine sum, not a product of diagonals.

6. Q6 of 6Hard

   Assertion (A): If A and B are invertible matrices of the same order, then (AB)⁻¹ = A⁻¹B⁻¹. Reason (R): The inverse of a square matrix, if it exists, is unique.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: A is false: Theorem 4 gives (AB)⁻¹ = B⁻¹A⁻¹, and since matrix multiplication is not commutative, A⁻¹B⁻¹ is in general a different matrix. R is true by Theorem 3.

Matrices notes: key concepts (Class 12 Mathematics)

A matrix is a rectangular array of numbers indexed by rows and columns. This chapter defines the standard types, the algebra of addition, scalar multiplication and non-commutative multiplication, the transpose and its laws, symmetric and skew-symmetric matrices, and invertibility — the toolkit determinants build on next.

Order of a matrix

An m × n matrix has m rows and n columns; the element aᵢⱼ sits in row i, column j. Order fixes which operations are allowed.

Types of matrices

Row (1 row), column (1 column), square (m = n), diagonal (off-diagonal zero), scalar (kI), identity (1's on diagonal) and zero matrices.

Matrix addition

Only matrices of the SAME order can be added, element by element; addition is commutative and associative.

Scalar multiplication

kA multiplies every entry of A by k; distributes over matrix addition, k(A + B) = kA + kB.

Matrix multiplication

AB is defined only when columns of A equal rows of B; (m×n)(n×p) gives m×p. It is associative but generally NOT commutative.

Transpose

Aᵀ interchanges rows and columns; (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ and (AB)ᵀ = BᵀAᵀ (order reverses).

Symmetric matrix

A square matrix with Aᵀ = A, i.e. aᵢⱼ = aⱼᵢ; it is mirror-symmetric across the main diagonal.

Skew-symmetric matrix

Aᵀ = −A, so aᵢⱼ = −aⱼᵢ; every diagonal entry is forced to be zero.

Symmetric–skew decomposition

Any square matrix splits uniquely as A = ½(A + Aᵀ) + ½(A − Aᵀ), a symmetric plus a skew-symmetric part.

Identity matrix

I has 1's on the diagonal and 0 elsewhere; AI = IA = A, so it is the multiplicative identity for square matrices.

Invertible matrix

A square A is invertible if AB = BA = I for some B; the inverse is unique and (AB)⁻¹ = B⁻¹A⁻¹.

Trace

The trace of a square matrix is the sum of its diagonal entries; tr(A + B) = tr A + tr B and tr(AB) = tr(BA).

Special square matrices

Idempotent (A² = A), nilpotent (Aⁿ = 0), orthogonal (AAᵀ = I) and involutory (A² = I) matrices appear in objective questions.