If [tex]$\begin{aligned} A & =\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \text { find the value of } \\ & -4 A+5 A\end{aligned}$[/tex]
Answer (2)
To solve the problem of finding the value of − 4 A + 5 A for the 2x2 matrix A , let's first understand the operations being performed:
Matrix A is defined as: A = [ a c b d ]
We are asked to find − 4 A + 5 A . This involves two operations on the matrix:
Scalar Multiplication:
Multiply matrix A by − 4 : − 4 A = − 4 × [ a c b d ] = [ − 4 a − 4 c − 4 b − 4 d ]
Scalar Multiplication:
Multiply matrix A by 5 : 5 A = 5 × [ a c b d ] = [ 5 a 5 c 5 b 5 d ]
Matrix Addition:
Add the two resulting matrices from the operations above: − 4 A + 5 A = [ − 4 a − 4 c − 4 b − 4 d ] + [ 5 a 5 c 5 b 5 d ]
The matrix addition is performed element-wise: = [ ( − 4 a + 5 a ) ( − 4 c + 5 c ) ( − 4 b + 5 b ) ( − 4 d + 5 d ) ] = [ a c b d ]
Upon performing these steps, we see that the resulting matrix is the same as the original matrix A . This demonstrates that when you multiply a matrix by a scalar and then add another scalar multiplication of the same matrix, it effectively combines them.
The value of − 4 A + 5 A simplifies to the original matrix A . This is because the operations of scalar multiplication and subsequent addition lead back to the original matrix. Therefore, the final result is A = [ a c b d ] .
;
