Let A=\left[\begin{array}{cc}2 & 1 \ 0 & -2]\end{array}, B=\left[\begin{array}{cc}4 & 1 \ -3 & -2]\end{array} and C=\left[\begin{array}{cc}-3 & 2 \ -1 & 4]\end{array} Find A^2+A C-5 B.

ICSE Class X Math 2014 Solved Paper - Question 37

Question : 37 of 52

Marks: +1, -0

Let

A={[\table 2 , 1 ; 0 , -2]}, B={[\table 4 , 1 ; -3 , -2]}

and

C={[\table -3 , 2 ; -1 , 4]}

Find

A^2+A C-5 B

Solution:

Given:

A={[\table 2 , 1 ; 0 , -2 ]}, {\B}\;={[\table 4 , 1 ; -3 , -2 ]},

{\C}={[\table -3 , 2 ; -1 , 4 ]}

{\A}^2= {\A} ⋅ {\A}\;={[\table 2 , 1 ; 0 , -2 ]}{[\table 2 , 1 ; 0 , -2 ]}

\;={[\table 4+0 , 2-2 ; 0+0 , 0+4 ]}

\;={[\table 4 , 0 ; 0 , 4 ]}

{\AC}\;={[\table 2 , 1 ; 0 , -2 ]}{[\table -3 , 2 ; -1 , 4 ]}

\;={[\table -6-1 , 4+4 ; 0+2 , 0-8 ]}={[\table -7 , 8 ; 2 , -8 ]}

and

5 B\;=5{[\table 4 , 1 ; -3 , -2 ]}

\;={[\table 20 , 5 ; -15 , -10 ]}

Now,

\;A^2+A C-5 B={[\table 4 , 0 ; 0 , 4 ]}

+{[\table -7 , 8 ; 2 , -8 ]}

\;\; -{[\table 20 , 5 ; -15 , -10 ]}

\;\; ={[\table 4-7-20 , 0+8-5 ; 0+2+15 , 4-8+10 ]}

\;\; ={[\table -23 , 3 ; 17 , 6 ]}

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