Determinants

What you'll learn

Learn what a determinant means, evaluate 2x2 and 3x3 determinants using cofactors, use the key properties as shortcuts, build the adjoint and inverse of a matrix, and solve linear equations with the matrix method and Cramer's Rule - all with fully worked steps for CBSE Class 12 and your JEE/NEET/CUET foundation.

This chapter has

Determinants — solved practice questions

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

1. Q1easy

   The value of the determinant | [[2, 4], [5, 3]] | is:

   • A-14
   • B14
   • C26
   • D-26
   Show answer & solution

   Correct answer: (A) -14

   For a 2 x 2 determinant, value = (2)(3) - (4)(5) = 6 - 20 = -14.

2. Q2easy

   For any square matrix A, the determinant of its transpose |A^T| equals:

   • A|A|
   • B-|A|
   • C1/|A|
   • D0
   Show answer & solution

   Correct answer: (A) |A|

   A determinant is unchanged when rows and columns are interchanged, so |A^T| = |A|.

3. Q3easy

   The value of the determinant of the diagonal matrix diag(2, 3, 4) is:

   • A9
   • B0
   • C12
   • D24
   Show answer & solution

   Correct answer: (D) 24

   The determinant of a diagonal (or triangular) matrix is the product of its diagonal entries: 2 x 3 x 4 = 24.

4. Q4easy

   A square matrix A is said to be singular if:

   • A|A| = 1
   • B|A| is not 0
   • C|A| = 0
   • DA is symmetric
   Show answer & solution

   Correct answer: (C) |A| = 0

   A matrix is singular precisely when its determinant is zero, in which case its inverse does not exist.

5. Q5medium

   If A is a 3 x 3 matrix with |A| = 5, then |2A| equals:

   • A10
   • B40
   • C80
   • D20
   Show answer & solution

   Correct answer: (B) 40

   For an n x n matrix, |kA| = k^n |A|. Here |2A| = 2^3 x 5 = 8 x 5 = 40.

6. Q6medium

   If A is a 3 x 3 matrix with |A| = 2, then |adj A| equals:

   • A2
   • B8
   • C16
   • D4
   Show answer & solution

   Correct answer: (D) 4

   For an n x n matrix, |adj A| = |A|^(n-1). Here |adj A| = 2^(3-1) = 2^2 = 4.

7. Q7medium

   For a square matrix A, the product A(adj A) equals:

   • A|A| I
   • BI
   • Cadj A
   • DO (zero matrix)
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   Correct answer: (A) |A| I

   By the fundamental adjoint property, A(adj A) = (adj A)A = |A| I.

8. Q8medium

   The matrix [[k, 2], [3, k]] has no inverse for:

   • Ak = plus or minus 2
   • Bk = plus or minus sqrt(6)
   • Ck = plus or minus 3
   • Dk = plus or minus 6
   Show answer & solution

   Correct answer: (B) k = plus or minus sqrt(6)

   No inverse means the matrix is singular, so k^2 - 6 = 0, giving k = plus or minus sqrt(6).

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