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Question: If \({X_{m \times 3}}{Y_{p \times 4}} = {Z_{2 \times b}}\) for three matrices \(X,Y\& Z\), then find...

If Xm×3Yp×4=Z2×b{X_{m \times 3}}{Y_{p \times 4}} = {Z_{2 \times b}} for three matrices X,Y&ZX,Y\& Z, then find the value of m×p×bm \times p \times b.

Explanation

Solution

Hint- In order to solve this question, we will use the property of multiplication of two matrices, which states that two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix and if the product is defined, the resulting matrix will have same number of rows as the first matrix and the same number of columns as the second matrix.

Complete step-by-step answer:
Given that: Xm×3Yp×4=Z2×b{X_{m \times 3}}{Y_{p \times 4}} = {Z_{2 \times b}}
We have to find m×p×bm \times p \times b
Here first matrix is X
Second matrix is Y
And resulting matrix is Z
As we know that two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.
So the number of column of first matrix = number of row of second matrix
∴3=p\therefore 3 = p or p=3p = 3
In this question, the product of the X and Y matrix is defined as the Z matrix.
We also know that if the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
According to this property
Row of first matrix = Row of resulting matrix
∴m=2\therefore m = 2
And the column of second matrix = Column of resulting matrix
∴4=b\therefore 4 = b or b=4b = 4
Therefore, b=4b = 4 , m=2m = 2 and p=3p = 3
We have to find term m×p×bm \times p \times b
Substituting the value of m, p and b we have
⇒m×p×b =2×3×4 =24  \Rightarrow m \times p \times b \\\ = 2 \times 3 \times 4 \\\ = 24 \\\
Hence the value of m×p×bm \times p \times b is 24.

Note- Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In order to solve such types of problems students must remember the relation between the number of rows and columns of a matrix before and after multiplication and what are the conditions for the multiplication of matrices.