Given:
The matrics
[tex]\begin{gathered} A=\begin{bmatrix}{a} & {2} & {3} \\ {4} & {b} & {6} \\ {c} & {8} & {-7}\end{bmatrix} \\ B=\begin{bmatrix}{5} & {1} & {-2} \\ {-1} & {3} & {4} \\ {2} & {d} & {e}\end{bmatrix} \\ AB=\begin{bmatrix}{14} & {-1} & {f} \\ {20} & {22} & {70} \\ {-27} & {44} & {-1}\end{bmatrix} \end{gathered}[/tex]Required:
Match lettered entries in the matrices A, B and AC to their values.
Explanation:
First multiply matrices A with B
[tex]\begin{gathered} AB=\begin{bmatrix}{a} & {2} & {3} \\ {4} & {b} & {6} \\ {c} & {8} & {-7}\end{bmatrix}\begin{bmatrix}{5} & {1} & {-2} \\ {-1} & {3} & {4} \\ {2} & {d} & {e}\end{bmatrix} \\ \begin{bmatrix}{5a-2+6} & {a+6+3d} & {-2a+8+3e} \\ {20-b+12} & {4+3b+6d} & {-8+4b+6e} \\ {5c-8-14} & {c+24-7d} & {-2c+32-7e}\end{bmatrix}=\begin{bmatrix}{14} & {-1} & {f} \\ {20} & {22} & {70} \\ {-27} & {44} & {-1}\end{bmatrix} \end{gathered}[/tex]Now we can find values of a, b, c, d, e and f from equating two matrices as:
[tex]\begin{gathered} 20-b+12=20 \\ b=12 \end{gathered}[/tex][tex]\begin{gathered} 5a-2+6=14 \\ a=2 \end{gathered}[/tex][tex]\begin{gathered} a+6+3d=-1 \\ 2+6+3d=-1 \\ 3d=-9 \\ d=-3 \end{gathered}[/tex][tex]\begin{gathered} -8+4b+6e=70 \\ -8+48+6e=70 \\ 6e=30 \\ e=5 \end{gathered}[/tex][tex]\begin{gathered} 5c-8-14=-27 \\ 5c=-5 \\ c=-1 \end{gathered}[/tex][tex]\begin{gathered} -2a+8+3e=f \\ -2(2)+8+3(5)=f \\ -4+8+15=f \\ 4+15=f \\ f=19 \end{gathered}[/tex]Answer:
The values are a = 2, b = 12, c = -1, d = -3, e = 5 and f = 19.
