Question

If \( A = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} \), then \( |A| \cdot \text{adj}(A) \) is equal to:

• A. \( a^{3n} \)
• B. \( a^{-3n} \)
• C. \( -a^{3n} \)
• D. \( 2a^{3n} \)

Hint:

For a diagonal matrix, the adjugate is easy to compute as it involves multiplying the identity matrix by powers of the diagonal elements.

Answer 1

The Correct Option is A

We are given the matrix \( A = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} \), and we are asked to find the value of \( |A| \cdot \text{adj}(A) \), where \( |A| \) denotes the determinant of \( A \) and \( \text{adj}(A) \) denotes the adjugate (or adjoint) of \( A \).

Step 1: Find the determinant of \( A \)

Since \( A \) is a diagonal matrix, the determinant \( |A| \) is simply the product of the diagonal elements: \[ |A| = a \cdot a \cdot a = a^3 \]

Step 2: Find the adjugate of \( A \)

The adjugate (or adjoint) of a matrix is the transpose of its cofactor matrix. For a diagonal matrix, the cofactor of each diagonal element is the product of the other diagonal elements. Hence, the cofactor matrix of \( A \) is: \[ \text{Cofactor}(A) = \begin{bmatrix} a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2 \end{bmatrix} \] Taking the transpose of this matrix gives the adjugate of \( A \): \[ \text{adj}(A) = \begin{bmatrix} a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2 \end{bmatrix} \]

Step 3: Multiply \( |A| \) with \( \text{adj}(A) \)

Now, we multiply the determinant \( |A| = a^3 \) by the adjugate \( \text{adj}(A) \): \[ |A| \cdot \text{adj}(A) = a^3 \cdot \begin{bmatrix} a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2 \end{bmatrix} \] This gives: \[ |A| \cdot \text{adj}(A) = \begin{bmatrix} a^5 & 0 & 0 \\ 0 & a^5 & 0 \\ 0 & 0 & a^5 \end{bmatrix} \]

Final Answer:

The matrix \( |A| \cdot \text{adj}(A) \) is equal to \( \boxed{\begin{bmatrix} a^5 & 0 & 0 \\ 0 & a^5 & 0 \\ 0 & 0 & a^5 \end{bmatrix}} \).