How do you solve a Matrix Equation?

Question

[8 7]X= [3 -6]
[1 1] [-2 9]

Just tell me how to do it plz

Answer 1

To solve a matrix equation of the form Ax = B, you isolate x by taking the inverse of A. Thus, you have:

x = A^-1*B

In the above equation, we have:

A =

[8 7]
[1 1]

B =

[3 -6]
[-2 9]

In general, for a given 2x2 matrix A, you obtain the inverse, A^-1 as follows. If A is defined as:

A =

[a b]
[c d]

then A^-1 is defined as:

[d -b]
[-c a]
multiplied by 1/det(A) where det(A) = a*d - b*c

For the given A above, then:

A^-1 =

[1 -7]
[-1 8]

multiplied by 1/det(A) = 1/(8*1 - 7*1) = 1

Thus you get:

A^-1 =

[1 -7]
[-1 8]

now solve for x = A^-1*B

x =

[(1)(3)+(-7)(-2) |||||||||| (1)(-6)+(-7)(9)]
[(-1)(3)+(8)(-2) |||||||||| (-1)(-6) + (8)(9)]

Solving, you get:

x =

[17 -69]
[-19 78]

To verify this answer, solve the equation Ax = B

A*x =

[(8)(17)+(7)(-19) |||||||||| (8)(-69)+(7)(78)]
[(1)(17)+(1)(-19) |||||||||| (1)(-69)+(1)(78)]

=

[136-133 |||||||||| -552+546]
[17-19 ||||||||||||||||||||| -69+78]

=

[3 -6]
[-2 9]

which is equivalent to B, thus verifying our answer for 'x'

--------

Hope this helps

Answer 2

Assuming this involves 2x2 matrices, note how X will be a matrix. If you know the identity matrix [1 0 , 0 1] when multiplied by a matrix A, gives A again you will see that all you need to find is such a matrix that when multiplied by the coefficient matrix of X, namely [8 7 , 1 1], gives the identity matrix as the coefficient matrix of X.
This matrix is the inverse of [8 7 , 1 1].

Answer 3

If row 1 of X is (a c) and row 2 of X is ( b d) then you multiply "row by column" to get two systems of equations.

For the first system

8 a + 7b = 3
a + b = -2

Substitute b = -2-a into the first equation and get a = 17, b = -19.

For the second system

8c + 7d = 6
c + d = 9

Substitute c = 9 - d into the first equation and get c = -69, d = 78.

Answer 4

One way is to multiply both sides by the inverse of
|8 7|
|1 1|

Answer 5

IF AX = B then

A^(-1)AX =A^(-1)(B)

ie X = A^(-1)(B)

Now A =

(8 7)
(1 1)

So A^(-1) =

.............(1 -7)
1/detA*
............(-1 8)

Since det A = 1 (= 1 *8 - -1 * -7) then 1/detA = 1

and so A^(-1) =

(1 -7)
(-1 8)

So X =

(1 -7) . (3 -6)
(......) * (......) =
(-1 8) . (-2 9)

(17 -69)
(-19 78)