Solve the matrix equation for $a,\ b,\ c,$a, b, c, and $d$d .$-3\left(\begin{bmatrix}7&2b\\8c&8\end{bmatrix}-\begin{bmatrix}4&4b\\5&d\end{bmatrix}\right)=\begin{bmatrix}a&54\\63&d\end{bmatrix}$−3([7 2b8c 8]−[4 4b5 d])=[a 5463 d]$a=$a= , $b=$b= , $c=$c= , $d=$d=
Question
Solve the matrix equation for a, b, c, and d .−3([7 2b8c 8]−[4 4b5 d])=[a 5463 d]a= , b= , c= , d=
Solution
The first step is to distribute the -3 into both matrices on the left side of the equation:
-3 * [7 2b; 8c 8] = [-21 -6b; -24c -24] -3 * [4 4b; 5 d] = [-12 -12b; -15 -3d]
Then, subtract the second matrix from the first:
[-21 -6b; -24c -24] - [-12 -12b; -15 -3d] = [-21+12 -6b+12b; -24c+15 -24+3d]
Simplify to get:
[-9 6b; -24c+15 -24+3d]
This matrix is equal to the matrix on the right side of the equation:
[-9 6b; -24c+15 -24+3d] = [a 54; 63 d]
From this, we can create a system of equations:
-9 = a 6b = 54 -24c + 15 = 63 -24 + 3d = d
Solving these equations gives:
a = -9 b = 54/6 = 9 c = (63 - 15) / -24 = -2 d = 24 / 3 = 8
So, the solution to the matrix equation is a = -9, b = 9, c = -2, d = 8.
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