A=⎣⎡023430011⎦⎤ B=⎣⎡11201335−1⎦⎤
(7.1) We know that dimension of matrix A=3X3 and dimension of matrix B = 3X3
Then, dimension of matrix AB will be 3X3
AB=⎣⎡0+4+02+3+23+0+20+4+00+3+30+0+30+20+06+15−19+0−1⎦⎤
AB=⎣⎡47546320208⎦⎤
(7.2) We know that if A and B are n x n matrices, then
det(AB) = (det A)(det B)
Proof:-
AB=⎣⎡47546320208⎦⎤
det(AB)=4(48−60)−4(56−100)+20(21−30)det(AB)=−52 ......(1)
and we know that
A=⎣⎡023430011⎦⎤ B=⎣⎡11201335−1⎦⎤
det(A)=0(3−0)−4(2−3)+0(0−9)det(A)=4
det(B)=1(−1−15)−0(−1−10)+3(3−2)det(B)=−13
and
det(A)×det(B)=4×(−13)=−52 .....(2)
So, from equation (1) and (2) , It is clear that
det(AB)=det(A)×det(B)
(7.3)
A=⎣⎡023430011⎦⎤ B=⎣⎡11201335−1⎦⎤
A+B=⎣⎡0+12+13+24+03+10+30+31+51−1⎦⎤
A+B=⎣⎡135443350⎦⎤
det(A+B)=1(0−15)−4(0−25)+3(9−20)det(A+B)=52
and
det(A)=0(3−0)−4(2−3)+0(0−9)det(A)=4
det(B)=1(−1−15)−0(−1−10)+3(3−2)det(B)=−13
So,
det(A)+det(B)=−9
So, we can say that det(A+B)=det(A)+det(B)
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