Matrices

What you'll learn

A from-scratch guide to Class 12 Matrices — what a matrix is, its order and types, how to add, scale, multiply and transpose them, and how symmetric and skew-symmetric matrices work, with every step shown.

This chapter has

Matrices — solved practice questions

8 Class 12 Mathematics questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1easy

   The number of possible orders of a matrix having exactly 24 elements is:

   • A6
   • B12
   • C24
   • D8
   Show answer & solution

   Correct answer: (D) 8

   Each order corresponds to a factor pair of 24: (1,24),(2,12),(3,8),(4,6),(6,4),(8,3),(12,2),(24,1). That gives 8 possible orders.

2. Q2easy

   If [[a, 3], [2, d]] = [[1, 3], [2, 4]], then the values of a and d are:

   • Aa = 3, d = 2
   • Ba = 1, d = 4
   • Ca = 4, d = 1
   • Da = 2, d = 3
   Show answer & solution

   Correct answer: (B) a = 1, d = 4

   Two matrices are equal only when corresponding entries are equal, so a = 1 and d = 4.

3. Q3easy

   A square matrix that is both symmetric and skew-symmetric must be:

   • AZero matrix
   • BIdentity matrix
   • CDiagonal matrix
   • DScalar matrix
   Show answer & solution

   Correct answer: (A) Zero matrix

   Symmetric means A = A^T and skew-symmetric means A^T = -A, so A = -A, giving 2A = O. Hence A is the zero (null) matrix.

4. Q4easy

   For two matrices A and B for which the product AB is defined, (AB)^T equals:

   • AA^T B^T
   • BB^T A^T
   • CAB
   • DBA
   Show answer & solution

   Correct answer: (B) B^T A^T

   By the reversal law for transpose of a product, (AB)^T = B^T A^T.

5. Q5medium

   The number of independent entries needed to specify a 3 x 3 symmetric matrix is:

   • A3
   • B9
   • C6
   • D5
   Show answer & solution

   Correct answer: (C) 6

   A symmetric matrix is determined by its diagonal and upper-triangular entries, that is n(n+1)/2 = 3(4)/2 = 6 entries.

6. Q6medium

   If A = [[0, 1], [0, 0]], then A^2 equals:

   • AA
   • BI
   • CNull (zero) matrix
   • D-A
   Show answer & solution

   Correct answer: (C) Null (zero) matrix

   A^2 = A x A = [[0*0+1*0, 0*1+1*0],[0,0]] = [[0,0],[0,0]], the null matrix.

7. Q7medium

   Every square matrix A can be written as the sum of a symmetric and a skew-symmetric matrix. The symmetric part is:

   • A(1/2)(A + A^T)
   • B(1/2)(A - A^T)
   • CA A^T
   • DA + A^T
   Show answer & solution

   Correct answer: (A) (1/2)(A + A^T)

   The standard decomposition uses symmetric part (1/2)(A + A^T) and skew-symmetric part (1/2)(A - A^T).

8. Q8medium

   If A is a matrix of order 3 x 2 and B is a matrix of order 2 x 4, then the order of the product AB is:

   • A2 x 4
   • B3 x 2
   • C2 x 2
   • D3 x 4
   Show answer & solution

   Correct answer: (D) 3 x 4

   When multiplying a 3 x 2 matrix by a 2 x 4 matrix, the inner dimensions match and the result has the outer dimensions, so AB is 3 x 4.

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