To find the value of \(|5A|\) when \(A\) is a \(2 \times 2\) matrix and \(|A| = 5\), we need to use a fundamental property of determinants related to scalar multiplication.

Determinant Property for Scalar Multiplication

For any square matrix \(A\) of order \(n \times n\) and any scalar \(k\), the determinant of the scalar multiple \(kA\) is given by the following property:

\[|kA| = k^n |A|\]

• The term \(|kA|\) represents the determinant of the matrix \(A\) after it has been multiplied by the scalar \(k\).
• \(k\) is the specific scalar value by which the matrix is multiplied. In this question, the scalar is \(5\).
• \(n\) is the order (or dimension) of the square matrix \(A\). For a \(2 \times 2\) matrix, \(n = 2\).
• \(|A|\) is the determinant of the original matrix \(A\), which is given as \(5\) in this problem.

Matrix A and Given Values

Let's identify the specific values provided in the problem for calculating the determinant \(|5A|\):

• The matrix \(A\) is stated to be a \(2 \times 2\) matrix. This directly tells us that the order of the matrix, \(n\), is \(2\).
• The determinant of matrix \(A\) is given as \(|A| = 5\).
• We are asked to find \(|5A|\), which means the scalar multiplier \(k\) is \(5\).

Calculating |5A| Using the Determinant Property

Now, we will substitute these identified values into the determinant property formula for scalar multiplication:

\[|kA| = k^n |A|\]

Substitute \(k = 5\), \(n = 2\), and \(|A| = 5\) into the formula:

\[|5A| = 5^2 \times |A|\]

First, calculate \(5^2\):

\[5^2 = 25\]

Now, substitute this value back into the equation:

\[|5A| = 25 \times 5\]

Perform the final multiplication:

\[|5A| = 125\]

Thus, the determinant of \(5A\) is \(125\). This calculation directly applies the determinant property for scalar multiples of matrices based on their dimension.