Let A = $[a_{ij}]_{2 imes 3}$ and B = $[b_{ij}]_{3 imes 2}$, then $|5AB|$ is equal to

Question

Let A = $[a_{ij}]_{2 	imes 3}$ and B = $[b_{ij}]_{3 	imes 2}$, then $|5AB|$ is equal to

Prepp Team · Accepted Answer

This question involves understanding the properties of determinants when dealing with matrix multiplication and scalar multiplication.

Determinant of Matrix Product

We are given two matrices, A and B, with the following dimensions:

Matrix A: $[a_{ij}]_{2 \times 3}$ (2 rows, 3 columns)
Matrix B: $[b_{ij}]_{3 \times 2}$ (3 rows, 2 columns)

The product of these two matrices, AB, is calculated as follows:

Dimension of A = $2 \times 3$

Dimension of B = $3 \times 2$

Dimension of AB = (Dimension of A) $\times$ (Dimension of B) = $(2 \times 3) \times (3 \times 2) = 2 \times 2$.

So, the resulting matrix AB is a $2 \times 2$ square matrix. Let's denote the product matrix as C = AB. Thus, C is a $2 \times 2$ matrix.

Scalar Multiplication Property of Determinants

We need to find the value of $|5AB|$. This means we need to find the determinant of the matrix obtained by multiplying the matrix AB by the scalar value 5.

A key property of determinants states that for any $n \times n$ square matrix M and any scalar k, the determinant of kM is given by:

$ |kM| = k^n |M| $

In our case:

The matrix is AB (or C), which is a $2 \times 2$ matrix.
The scalar multiple is k = 5.
The dimension of the square matrix AB is $n=2$.

Applying the property, we get:

$ |5AB| = 5^n |AB| $

Substituting $n=2$:

$ |5AB| = 5^2 |AB| $

Addressing Other Options

It's important to note why other options are incorrect:

Options 1 and 2 ($5^2. |A|. |B|$ and $5^3. |A|. |B|$) are incorrect because the determinants $|A|$ and $|B|$ are not defined. The determinant is only defined for square matrices, and matrices A ($2 \times 3$) and B ($3 \times 2$) are rectangular matrices. Furthermore, the property $|AB| = |A||B|$ only applies when both A and B are square matrices of the same dimension.
Option 4 ($5^3|AB|$) uses the wrong power of the scalar. The power corresponds to the dimension of the resulting square matrix after multiplication, which is 2 in this case, not 3.