what is the solution of the martix equation? [9/2 -5/-1] X=[8/-7 5/1]

Question

Im in honors algebra 2.

Answer 1

This question has not used the standard notation in expressing these matrices, so I've made certain assumptions. Nevertheless, the method applies as shown.

Let X=

⎡x₁₁ x₁₂⎤

⎣x₂₁ x₂₂⎦

I have assumed that A=

⎡ 9  2⎤

⎣-5 -1⎦

I have assumed that B=

⎡ 8 -7⎤

⎣ 5  1⎦

then AX=B.

A^-1A=I where I is the identity matrix:

⎡ 1  0⎤

⎣ 0  1⎦

Let A^-1=

⎡a b⎤

⎣c d⎦

So we can write the following 4 equations:

9a-5b=1, 2a-b=0, so b=2a, 9a-10a=1, a=-1, b=-2;

9c-5d=0, 2c-d=1, so d=2c-1, 9c-10c+5=0, c=5, d=9.

A^-1=

⎡-1 -2⎤

⎣ 5  9⎦

A^-1AX=A^-1B,

X=A^-1B.

x₁₁=-8-10=-18, x₁₂=7-2=5, x₂₁=40+45=85, x₂₂=-35+9=-26.

X=

⎡-18    5⎤

⎣ 85 -26⎦

CHECK

AX=

⎡ 9  2⎤⎡-18    5⎤

⎣-5 -1⎦⎣ 85 -26⎦=

⎡-162+170 45-52⎤

⎣ 90-85    -25+26⎦=

⎡ 8 -7⎤

⎣ 5  1⎦= B

An alternative method is to bypass the inverse matrix method and simply solve the following be replacing X with:

⎡a b⎤

⎣c d⎦:

9a+2c=8, -5a-c=5⇒a=-18, c=85;

9b+2d=-7, -5b-d=1⇒b=5, d=-26.

If my assumptions about A and B are not as intended by the question then follow the method with the appropriate switching of rows and columns. This will produce X=

⎡-43  0⎤

⎣-79 -1⎦