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.
Conclusion
Based on the property of scalar multiplication of determinants for square matrices, the correct expression for $|5AB|$ is $5^2 |AB|$.
Important Questions from Matrix Algebra
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
If A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), then which of the following statements are correct?
A. A is a square matrix
B. A−1 exists
C. A is a symmetric matrix
D. |A| = 19
E. A is a null matrix
Choose the correct answer from the options given below.
If A is Square Matrix of order 3, then product of A and its transpose is
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&1&0\\ 0&1&0\\ 1&1&0\\ 0&0&1 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 1&0&0&0\\ 0&1&1&0\\ 1&0&1&1\\ \end{array}} \right]\) Find the boolean product A ⊙ B of the two matrices.
