Question

If A is a \( 3 \times 3 \) matrix, \( |A| \neq 0 \) and \( |3A| = k |A| \), then the values of k is:

• A. \( 3 \)
• B. \( 9 \)
• C. \( 8 \)
• D. \( 27 \)

Hint:

Remember the fundamental property of determinants: If A is an \( n \times n \) matrix and c is a scalar, then \( \det(cA) = c^n \det(A) \). The scalar gets raised to the power of the matrix dimension.

Answer 1

The Correct Option is D

Step 1: Recall the property of determinants involving scalar multiplication. For any \( n \times n \) matrix A and any scalar \( c \), the determinant of the matrix \( cA \) is given by: \[ \det(cA) = c^n \det(A) \] or using the notation from the question: \[ |cA| = c^n |A| \] Step 2: Identify the values of \( n \) and the scalar \( c \) in this problem. The matrix A is a \( 3 \times 3 \) matrix, so \( n = 3 \). The scalar multiple is 3, so \( c = 3 \). Step 3: Apply the property to the specific case \( |3A| \). Using the formula with \( n=3 \) and \( c=3 \): \[ |3A| = 3^3 |A| \] \[ |3A| = 27 |A| \] Step 4: Compare this result with the given equation \( |3A| = k |A| \). We have: \[ 27 |A| = k |A| \] Step 5: Solve for k. Since we are given that \( |A| \neq 0 \), we can divide both sides of the equation by \( |A| \): \[ 27 = k \] So, the value of k is 27. Step 6: Compare the result with the given options. The calculated value \( k = 27 \) matches option (D).