Question

If A=$\begin{bmatrix} 3 & 0 & 0 \\ 2 & 2 & 0 \\ 4 & 5 & 3 \end{bmatrix}$ then which of the following is (are) true?

• A. A is invertible.
• B. trace (adj (adj A))=144.
• C. trace (adj (adj A))= 108.
• D. |adj A| is less than 400.

Answer 1

Answer: (A), (B), (D)

(A), (B), (D)

Answer 2

Let the given matrix be $A = \begin{bmatrix} 3 & 0 & 0 \\ 2 & 2 & 0 \\ 4 & 5 & 3 \end{bmatrix}$. The order of the matrix A is $n=3$.

We analyze each statement:

(A) A is invertible.

A matrix A is invertible if and only if its determinant is non-zero. The given matrix A is a lower triangular matrix. The determinant of a triangular matrix is the product of its diagonal elements. $|A| = \det(A) = 3 \times 2 \times 3 = 18$. Since $|A| = 18 \neq 0$, the matrix A is invertible. So, statement (A) is true.

(B) trace (adj (adj A))=144.

(C) trace (adj (adj A))= 108.

For a non-singular matrix A of order n, the adjugate of the adjugate of A is given by the property: $\text{adj}(\text{adj } A) = |A|^{n-2} A$. In this case, $n=3$ and $|A|=18$. Since $|A| \neq 0$, A is non-singular. $\text{adj}(\text{adj } A) = |A|^{3-2} A = |A|^1 A = |A| A = 18A$. The trace of a matrix is the sum of its diagonal elements. The trace of a scalar multiple of a matrix is the scalar multiple of the trace. $\text{trace}(kA) = k \text{ trace}(A)$. $\text{trace}(\text{adj}(\text{adj } A)) = \text{trace}(18A) = 18 \text{ trace}(A)$. The trace of A is the sum of its diagonal elements: $\text{trace}(A) = 3 + 2 + 3 = 8$. So, $\text{trace}(\text{adj}(\text{adj } A)) = 18 \times 8 = 144$. Statement (B) says trace (adj (adj A))=144, which is true. Statement (C) says trace (adj (adj A))= 108, which is false.

(D) |adj A| is less than 400.

For a non-singular matrix A of order n, the determinant of the adjugate of A is given by the property: $|\text{adj } A| = |A|^{n-1}$. In this case, $n=3$ and $|A|=18$. $|\text{adj } A| = |A|^{3-1} = |A|^2 = 18^2$. $18^2 = 324$. The statement says |adj A| is less than 400. $324 < 400$. This inequality is true. So, statement (D) is true.

Based on the analysis, statements (A), (B), and (D) are true.