Find the inverses of the matrices.
Let A = 1 2
5 12
b1 = -1
3
b2 = 1
-5
b3 = 2
6
b4 = 3
5
Find A^-1, and use it to solve the four equations.
Ax = b1, Ax = b2, Ax = b3, Ax = b4
Only A has an inverse. Only square matrices have inverses. A^-1 =
6 -1
-5/2 1/2
Now just plug it in
To find the inverse of a matrix, you can follow these steps:
1. Write down the given matrix A.
A = 1 2
5 12
2. Set up an augmented matrix [A | I] where I is the identity matrix of the same size as A.
[A | I] = 1 2 | 1 0
5 12 | 0 1
3. Perform elementary row operations to transform the left side of the augmented matrix into the identity matrix. Keep track of the row operations you used.
[A | I] = 1 2 | 1 0 (R1)
5 12 | 0 1 (R2)
Multiply R1 by 5 and subtract R2 multiplied by 1 to eliminate the entry in the lower-left corner:
[A | I] = 1 2 | 1 0 (R1)
0 2 | -5 1 (R2)
Divide R2 by 2 to make the entry in the bottom-right corner equal to 1:
[A | I] = 1 2 | 1 0 (R1)
0 1 | -2.5 0.5 (R2)
Subtract 2 times R2 from R1 to eliminate the entry in the top-right corner:
[A | I] = 1 0 | 6.5 -1
0 1 | -2.5 0.5
The left side is now the identity matrix, so the right side contains the inverse of A:
A^-1 = 6.5 -1
-2.5 0.5
To solve the equations Ax = b1, Ax = b2, Ax = b3, and Ax = b4, you can multiply both sides of each equation by A^-1:
A^-1(Ax) = A^-1(b1)
x = A^-1(b1)
A^-1(Ax) = A^-1(b2)
x = A^-1(b2)
A^-1(Ax) = A^-1(b3)
x = A^-1(b3)
A^-1(Ax) = A^-1(b4)
x = A^-1(b4)
Substitute the values of b1, b2, b3, and b4 into the equations and multiply them by A^-1 to get the solutions for x.