Question

If A is a square matrix of order 4 and |A| = 4, then |2A| will be:

• A. 8
• B. 64
• C. 16
• D. 4

Hint:

When working with the determinant of a scalar multiple of a matrix, the key idea is that multiplying a matrix \(A\) by a scalar \(k\) results in the determinant being multiplied by \(k^n\), where \(n\) is the size of the matrix. This property helps simplify the calculation of determinants when the matrix is scaled by a constant. For a \(4 \times 4\) matrix, raising the scalar to the power of 4 is crucial for finding the correct result.

Answer 1

The Correct Option is B

To determine the value of |2A| given that A is a square matrix of order 4 and |A|=4, we can use the property of determinants which states that for any square matrix A of order n and scalar k, the determinant of the scaled matrix kA is given by:

$\left|\mathrm{k}A\right|=\left|k\right|^\mathrm{n}\left|A\right|$

Here, k = 2 and n = 4. Therefore, we calculate:

$\left|2A\right|=2^4\left|A\right|$

Substituting the known values:

$\left|2A\right|=16\times 4=64$

Therefore, the value of |2A| is 64.

Answer 2

For an \(n \times n\) matrix, the determinant of a scalar multiple of the matrix is given by:

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

Here, we are given that \(A\) is a \(4 \times 4\) matrix and we are asked to find \( |2A| \). Applying the formula:

\[ |2A| = 2^4 \cdot |A| \]

The determinant of \(2A\) is then:

\[ |2A| = 2^4 \cdot 4 = 16 \cdot 4 = 64 \]

Conclusion: Thus, \( |2A| = 64 \).