# CBSE Class 12th Math 4 – Determinants MCQs

#### Which of the following is a correct statement?

Correct! Wrong!

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A where aij = (i, j)th element of A.

#### If A2 + A - I = 0, then A-1 is ____.

Correct! Wrong!

Given that A2 + A - I = 0 Premultiplying both sides by A-1, (A-1 A)A + A-1 A - A-1I = 0 ⇒ IA + I - A-1= 0 (since, A-1A = I) ⇒ A + I - A-1= 0 [IA = A] ⇒ A-1= A + I

#### If A is a square matrix of order 4 such that |adj A| = 125, then |A| is ____.

Correct! Wrong!

We know that |adj A| = |A|n-1, where n is the order of the matrix. Therefore 125 = |A|4 – 1 ⇒125 = A3 ⇒A = 5

#### A homogeneous system of n linear equations in n unknowns is expressible in the form AX = 0. The system has non-trivial solution if ____.

Correct! Wrong!

A homogeneous system of n linear equations in n unknowns is expressible in the form AX = 0. A homogeneous system of equation is always consistent. If |A| ≠ 0, then AX = 0 has unique solution. This solution is called the trivial solution, where x = y = z = 0. If |A| = 0, then AX = 0 has infinitely many solutions.

#### If A = [aij] is a square matrix of order n × n and k is a scalar then |kA| is equal to ____.

Correct! Wrong!

If A = kB where A and B are square matrices of order n, then |A| = kn |B| where n = 1, 2, 3.

Correct! Wrong!

#### If for matrix A, |A| = 3, where matrix A is of order 2 × 2, then |5 A| is ____.

Correct! Wrong!

|A| = 3 (given) |5 A| = 52 |A| [matrix A is of order 2 × 2] = 25 |A| = 25 × 3 = 75

#### A square matrix A of order n is invertible if there exists a square matrix B of the same order such that AB = In = BA. In such a case, we say that the inverse of matrix A is B. Following are some properties of inverse of a matrix. The property that is not true is ____.

Correct! Wrong!

If A is an invertible symmetric matrix, then AT = A ⇒ (AT)-1 = A-1 ⇒ (A-1)T = A-1 [since, (AT)-1 = (A-1)T] Therefore, the inverse of an invertible symmetric matrix is a symmetric matrix. Hence, the statement given in Option A is false.

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