Matrix Division Definition and Examples

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3. Matrix Division example ( Enter your problem )

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Definition and Examples
Example-2

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2. Multiplication of two matrix
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2. Example-2
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1. Definition and Examples

1. Division of matrix

1.

AB=A⋅B-1 $A/B=A*B^-1$
2.

BA=B⋅A-1 $B/A=B*A^-1$

Examples
1. Find BA $B/A$ ...
A=[231056112] $A=[[2,3,1],[0,5,6],[1,1,2]]$ ,B=[21-110-1112] $B=[[2,1,-1],[1,0,-1],[1,1,2]]$

Solution:

|A| $|A|$

=

	2 $2$	3 $3$	1 $1$
	0 $0$	5 $5$	6 $6$
	1 $1$	1 $1$	2 $2$

=

2 $2$

×

	5 $5$	6 $6$
	1 $1$	2 $2$

-3 $-3$

×

	0 $0$	6 $6$
	1 $1$	2 $2$

+1 $+1$

×

	0 $0$	5 $5$
	1 $1$	1 $1$

=2×(5×2-6×1)-3×(0×2-6×1)+1×(0×1-5×1) $=2 xx (5 × 2 - 6 × 1) -3 xx (0 × 2 - 6 × 1) +1 xx (0 × 1 - 5 × 1)$

=2×(10-6)-3×(0-6)+1×(0-5) $=2 xx (10 -6) -3 xx (0 -6) +1 xx (0 -5)$

=2×(4)-3×(-6)+1×(-5) $=2 xx (4) -3 xx (-6) +1 xx (-5)$

=8+18-5 $= 8 +18 -5$

=21 $=21$

Adj(A) $Adj(A)$

=

Adj

	2 $2$	3 $3$	1 $1$
	0 $0$	5 $5$	6 $6$
	1 $1$	1 $1$	2 $2$

=

+

	5 $5$	6 $6$
	1 $1$	2 $2$

-

	0 $0$	6 $6$
	1 $1$	2 $2$

+

	0 $0$	5 $5$
	1 $1$	1 $1$

-

	3 $3$	1 $1$
	1 $1$	2 $2$

+

	2 $2$	1 $1$
	1 $1$	2 $2$

-

	2 $2$	3 $3$
	1 $1$	1 $1$

+

	3 $3$	1 $1$
	5 $5$	6 $6$

-

	2 $2$	1 $1$
	0 $0$	6 $6$

+

	2 $2$	3 $3$
	0 $0$	5 $5$

^T

=

	+(5×2-6×1) $+(5 × 2 - 6 × 1)$	-(0×2-6×1) $-(0 × 2 - 6 × 1)$	+(0×1-5×1) $+(0 × 1 - 5 × 1)$
	-(3×2-1×1) $-(3 × 2 - 1 × 1)$	+(2×2-1×1) $+(2 × 2 - 1 × 1)$	-(2×1-3×1) $-(2 × 1 - 3 × 1)$
	+(3×6-1×5) $+(3 × 6 - 1 × 5)$	-(2×6-1×0) $-(2 × 6 - 1 × 0)$	+(2×5-3×0) $+(2 × 5 - 3 × 0)$

^T

=

	+(10-6) $+(10 -6)$	-(0-6) $-(0 -6)$	+(0-5) $+(0 -5)$
	-(6-1) $-(6 -1)$	+(4-1) $+(4 -1)$	-(2-3) $-(2 -3)$
	+(18-5) $+(18 -5)$	-(12+0) $-(12 +0)$	+(10+0) $+(10 +0)$

^T

=

	4 $4$	6 $6$	-5 $-5$
	-5 $-5$	3 $3$	1 $1$
	13 $13$	-12 $-12$	10 $10$

^T

=

	4 $4$	-5 $-5$	13 $13$
	6 $6$	3 $3$	-12 $-12$
	-5 $-5$	1 $1$	10 $10$

Now, A-1=1|A|×Adj(A) $"Now, "A^(-1)=1/|A| × Adj(A)$

=

121 $1/(21)$ ×

	4 $4$	-5 $-5$	13 $13$
	6 $6$	3 $3$	-12 $-12$
	-5 $-5$	1 $1$	10 $10$

=

	0.1905 $0.1905$	-0.2381 $-0.2381$	0.619 $0.619$
	0.2857 $0.2857$	0.1429 $0.1429$	-0.5714 $-0.5714$
	-0.2381 $-0.2381$	0.0476 $0.0476$	0.4762 $0.4762$

B×(A-1) $B×(A^(-1))$

=

	2 $2$	1 $1$	-1 $-1$
	1 $1$	0 $0$	-1 $-1$
	1 $1$	1 $1$	2 $2$

×

	0.1905 $0.1905$	-0.2381 $-0.2381$	0.619 $0.619$
	0.2857 $0.2857$	0.1429 $0.1429$	-0.5714 $-0.5714$
	-0.2381 $-0.2381$	0.0476 $0.0476$	0.4762 $0.4762$

=

	2×0.1905+1×0.2857-1×-0.2381 $2×0.1905+1×0.2857-1×-0.2381$	2×-0.2381+1×0.1429-1×0.0476 $2×-0.2381+1×0.1429-1×0.0476$	2×0.619+1×-0.5714-1×0.4762 $2×0.619+1×-0.5714-1×0.4762$
	1×0.1905+0×0.2857-1×-0.2381 $1×0.1905+0×0.2857-1×-0.2381$	1×-0.2381+0×0.1429-1×0.0476 $1×-0.2381+0×0.1429-1×0.0476$	1×0.619+0×-0.5714-1×0.4762 $1×0.619+0×-0.5714-1×0.4762$
	1×0.1905+1×0.2857+2×-0.2381 $1×0.1905+1×0.2857+2×-0.2381$	1×-0.2381+1×0.1429+2×0.0476 $1×-0.2381+1×0.1429+2×0.0476$	1×0.619+1×-0.5714+2×0.4762 $1×0.619+1×-0.5714+2×0.4762$

=

	0.381+0.2857+0.2381 $0.381+0.2857+0.2381$	-0.4762+0.1429-0.0476 $-0.4762+0.1429-0.0476$	1.2381-0.5714-0.4762 $1.2381-0.5714-0.4762$
	0.1905+0+0.2381 $0.1905+0+0.2381$	-0.2381+0-0.0476 $-0.2381+0-0.0476$	0.619+0-0.4762 $0.619+0-0.4762$
	0.1905+0.2857-0.4762 $0.1905+0.2857-0.4762$	-0.2381+0.1429+0.0952 $-0.2381+0.1429+0.0952$	0.619-0.5714+0.9524 $0.619-0.5714+0.9524$

=

	0.9048 $0.9048$	-0.381 $-0.381$	0.1905 $0.1905$
	0.4286 $0.4286$	-0.2857 $-0.2857$	0.1429 $0.1429$
	0 $0$	0 $0$	1 $1$

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2. Multiplication of two matrix
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2. Example-2
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