Use the given matrices to evaluate the expression.$A=\begin{bmatrix}4&1\\-3&0\end{bmatrix}$A=[4 1−3 0] , $B=\begin{bmatrix}5&-2\\6&-1\end{bmatrix}$B=[5 −26 −1] , $C=\begin{bmatrix}1&0\\0&\ \ 1\end{bmatrix}$C=[1 00 1]$AC+BC=$AC+BC=
Question
Use the given matrices to evaluate the expression.A=[4 1−3 0] , B=[5 −26 −1] , C=[1 00 1]AC+BC=
Solution
To evaluate the expression AC + BC, we first need to find the products AC and BC.
Matrix multiplication is done element by element, with the element in the i-th row and j-th column of the product being the sum of the product of the corresponding elements in the i-th row of the first matrix and the j-th column of the second matrix.
- Calculate AC:
A = [4 1; -3 0], C = [1 0; 0 1]
AC = [41 + 10, 40 + 11; -31 + 00, -30 + 01] = [4, 1; -3, 0]
- Calculate BC:
B = [5 -2; 6 -1], C = [1 0; 0 1]
BC = [51 + -20, 50 + -21; 61 + -10, 60 + -11] = [5, -2; 6, -1]
- Now, add the two resulting matrices (AC + BC):
AC + BC = [4 + 5, 1 + -2; -3 + 6, 0 + -1] = [9, -1; 3, -1]
So, AC + BC = [9 -1; 3 -1].
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