Question

If \(A=\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 3 \\\ 0 & 0 & 2\end{array}\right]\), then the determinant \(A B\) has the value (a) 4 (b) 8 (c) 16 (d) 32

Short answer

The determinant of AB is 16; hence, the answer is (c) 16.

Step-by-step solution

Determine the Determinant of Matrix A

Matrix \( A \) is a diagonal matrix with all diagonal entries equal to 2. The determinant of a diagonal matrix is the product of its diagonal elements. Hence, \( \det(A) = 2 \times 2 \times 2 = 8 \).

Determine the Determinant of Matrix B

Matrix \( B \) is an upper triangular matrix where the determinant is also the product of its diagonal elements. Thus, \( \det(B) = 1 \times 1 \times 2 = 2 \).

Use the Property of Determinants with Matrix Multiplication

For any two square matrices, the property \( \det(AB) = \det(A) \times \det(B) \) holds. Using this property, calculate \( \det(AB) \) as follows: \( \det(AB) = \det(A) \times \det(B) = 8 \times 2 = 16 \).

Verify and Conclude

Verify the calculated determinant value against the given options. The value of \( \det(AB) \) is 16, which corresponds to option (c).

Key concepts

Diagonal Matrix

A diagonal matrix is one where all the elements outside the main diagonal are zero. This means that if you look at a square matrix, only the elements from the top left to the bottom right are potentially non-zero. In simpler terms, it's a matrix that looks like a staircase. For example, our matrix \( A \), which is \( \begin{bmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2 \end{bmatrix} \), qualifies as a diagonal matrix because all the off-diagonal elements are zero.
An interesting property of diagonal matrices is calculating their determinant. This can be done by multiplying all the diagonal elements together. So for any \( n \times n \) diagonal matrix with diagonal elements \( d_1, d_2, \ldots, d_n \), the determinant is \( d_1 \times d_2 \times \ldots \times d_n \). For matrix \( A \), this translates to \( 2 \times 2 \times 2 = 8 \). This calculation is straightforward and demonstrates one of the conveniences of diagonal matrices.

Upper Triangular Matrix

When you see an upper triangular matrix, it's a matrix where all the elements below the diagonal are zero. Think of it as a triangle at the top of the matrix. Consider matrix \( B \), which is \( \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 3 \ 0 & 0 & 2 \end{bmatrix} \). Here, you will notice zeros positioned below the main diagonal from the upper left to the lower right.
Like diagonal matrices, upper triangular matrices have a simple formula for determining their determinant. The determinant is the product of the diagonal elements. Thus, for matrix \( B \), this is \( 1 \times 1 \times 2 = 2 \). This simple process reveals how the triangular structure aids in performing the determinant calculation with ease. It emphasizes the elegance of using triangular matrices in mathematical computations.

Matrix Multiplication

Matrix multiplication involves multiplying the rows of the first matrix with the columns of the second. However, what's intriguing is how this relates to determinants. This brings us to an important property: the determinant of a product of matrices is equal to the product of their determinants. Mathematically, this is denoted as \( \det(AB) = \det(A) \times \det(B) \).
In our exercise, both \( A \) and \( B \) are square matrices, so we can apply this property. By calculating, \( \det(A) = 8 \) and \( \det(B) = 2 \), you find \( \det(AB) = 8 \times 2 = 16 \). This principle significantly simplifies the computation and is incredibly useful in complex matrix calculations, emphasizing the efficiency of matrix algebra.

Determinant Properties

Determinants have various properties that can make solving matrix-related problems simpler. Here are a few crucial ones:

• The determinant of a diagonal or triangular matrix (upper or lower) is the product of its diagonal elements. This can greatly simplify calculations as seen in our matrices \( A \) and \( B \).
• When matrices are multiplied together, the determinant of the product equals the product of the determinants. So, \( \det(AB) = \det(A) \times \det(B) \), which we've applied successfully to find \( \det(AB) = 16 \).
• A matrix with a row or column of zeros has a determinant of zero, simplifying certain matrix evaluations drastically.
• The determinant changes sign if two rows or columns are swapped. This property plays a role in altering results in more complex operations.

Understanding these properties helps in efficiently navigating through matrix equations and reveals the interconnectedness of matrix algebra structures.