Matrix Adjoint Determinant Explanation

Understanding the properties of matrices, specifically the relationship between the determinant of a matrix and the determinant of its adjoint, is crucial in linear algebra. This problem involves a square matrix P of order 3, and we are given its determinant, |P|.

Given Matrix Information

• The matrix in question is a square matrix, denoted as P.
• The order of the matrix P (denoted as 'n') is 3. This means P is a 3x3 matrix.
• The determinant of matrix P, denoted as $$\left| P \right|$$, is given as -4.

Adjoint Determinant Formula

For any square matrix A of order 'n', the determinant of its adjoint, denoted as $$\left| \text{adj A} \right|$$, is related to the determinant of the matrix A itself by the following formula:

$$\left| \text{adj A} \right| = {\left| \text{A} \right|}^{n-1}$$

Where:

• $$\left| \text{adj A} \right|$$ represents the determinant of the adjoint of matrix A.
• $$\left| \text{A} \right|$$ represents the determinant of matrix A.
• n represents the order of the square matrix A.

Calculating Adjoint Determinant

Now, let's apply this formula to the given problem. We have matrix P, where:

• $n = 3$ (order of matrix P)
• $$\left| P \right| = -4$$ (determinant of matrix P)

Using the formula for the determinant of the adjoint matrix:

$$\left| \text{adj P} \right| = {\left| \text{P} \right|}^{n-1}$$

Substitute the given values into the formula:

$$\left| \text{adj P} \right| = {\left( -4 \right)}^{3-1}$$

$$\left| \text{adj P} \right| = {\left( -4 \right)}^{2}$$

Calculate the square of -4:

$$\left| \text{adj P} \right| = 16$$

Final Adjoint Determinant Value

Therefore, the determinant of the adjoint of matrix P, $$\left| \text{adj P} \right|$$, is 16.