Understanding the Matrix Problem: Finding the Determinant \(|A|\)

The problem asks us to find the value of the determinant of a matrix \( A \), denoted by \( |A| \), given the equation involving the matrix \( A \) and its adjoint \( \operatorname{adj} A \).

The given equation is:

\( A \operatorname{adj} A = \begin{bmatrix} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -5 \end{bmatrix} \)

Key Property of Adjoint Matrix and Determinant

A fundamental property of square matrices relates the matrix, its adjoint, and its determinant. For any square matrix \( A \) of order \( n \), the following relationship holds:

\( A (\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I_n \)

where \( |A| \) is the determinant of \( A \) and \( I_n \) is the identity matrix of order \( n \).

Analyzing the Given Matrix Equation

First, let's look at the matrix on the right-hand side of the given equation:

\( \begin{bmatrix} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -5 \end{bmatrix} \)

This is a diagonal matrix. We can factor out the scalar \( -5 \) from this matrix:

\( \begin{bmatrix} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -5 \end{bmatrix} = -5 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)

The matrix \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) is the identity matrix of order 3, denoted by \( I_3 \). This tells us that the original matrix \( A \) must be a 3x3 matrix, as the identity matrix \( I \) in the property \( A (\operatorname{adj} A) = |A| I \) must be of the same order as \( A \).

So, the given equation becomes:

\( A \operatorname{adj} A = -5 I_3 \)

Solving for the Determinant \(|A|\)

Now we can compare this equation with the fundamental property:

\( A (\operatorname{adj} A) = |A| I_n \)

Substituting \( n=3 \), we have:

\( A (\operatorname{adj} A) = |A| I_3 \)

By comparing \( A \operatorname{adj} A = -5 I_3 \) and \( A \operatorname{adj} A = |A| I_3 \), we can directly equate the scalar coefficients multiplying the identity matrix \( I_3 \).

Therefore, we get:

\( |A| I_3 = -5 I_3 \)

Since \( I_3 \) is a non-zero matrix, we can conclude that the scalar coefficients must be equal.

\( |A| = -5 \)

Conclusion on the Determinant Value

Based on the property \( A \operatorname{adj} A = |A| I \) and the given equation, the determinant of matrix \( A \) is found to be \( -5 \).

Revision Table: Key Matrix Concepts

Additional Information: Adjoint Matrix Properties and Inverse

The adjoint matrix is closely related to the inverse of a matrix. If a square matrix \( A \) is invertible (i.e., \( |A| \neq 0 \)), its inverse \( A^{-1} \) can be calculated using the adjoint matrix and the determinant:

\( A^{-1} = \frac{1}{|A|} \operatorname{adj} A \)

This formula comes directly from the property \( A (\operatorname{adj} A) = |A| I \). If \( |A| \neq 0 \), we can multiply both sides by \( \frac{1}{|A|} \):

\( \frac{1}{|A|} A (\operatorname{adj} A) = \frac{1}{|A|} (|A| I) \)

\( A \left( \frac{1}{|A|} \operatorname{adj} A \right) = I \)

By definition of matrix inverse, \( A A^{-1} = I \). Comparing this with the equation above, we see that \( A^{-1} = \frac{1}{|A|} \operatorname{adj} A \).

In our specific problem, since \( |A| = -5 \) (which is not zero), the matrix \( A \) is invertible. If \( |A| \) were 0, the matrix \( A \) would be singular and would not have an inverse.