Question

If A = `[(1,3), (3,4)]` B = `[(-2,1), (-3,2)]` and `A^2 - 5B^2 = 5C` Find the matrix C where C is a 2 by 2 matrix. 

Answer 1

A² − 5B² = 5C

`A = [(1,3),(3,4)], B = [(-2,1),(-3,2)]`

Step 1: Compute A²

`A^2 = A.A = [(1,3),(3,4)].[(1,3),(3,4)]`

Perform matrix multiplication:

`A^2 = [(1xx1 + 3xx3,1xx3+3xx4),(3xx1+4xx3,3xx3+4xx4)]`

`A^2 = [(1+9,3+12),(3+12,9+16)]`

`A^2 = [(10,15),(15,25)]`

Step 2: Compute B²

`B^2 = BxxB = [(-2,1),(-3,2)]xx[(-2,1),(-3,2)]`

Perform matrix multiplication:

`B^2 = [(-2xx-2+1xx-3,-2xx1+1xx2),(-3xx-2+2xx-3,-3xx1+2xx2)]`

`B^2 = [(4-3,-2+2),(6-6,-3+4)]`

`B^2 = [(1,0),(0,1)]`

Step 3: Compute 5B²

`5B^2 = 5xx[(1,0),(0,1)]`

`5B^2 = [(5,0),(0,5)]`

Step 4: Compute A²−5B²

`A^2 - 5B^2 = [(10,15),(15,25)]-[(5,0),(0,5)]`

`A^2 - 5B^2 = [(10-5,15-0),(15-0,25-5)]`

`A^2 - 5B^2 = [(5,15),(15,20)]`

Step 5: Solve for C

`A^2 - 5B^2 = 5C`

`5C = [(5,15),(15,20)]`

Divide each element by 5:

`C= [(1,3),(3,4)]`