# Solve for the matrix X in the following equation. [[2,1],[1,4]]X^T = -4[[4,1,0],[2,2,5]] + [ [0,0],[0,3]]X^T

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### 2 Answers

You need to select the matrix X as it follows:

`X = ((a,b),(c,d),(e,f)) `

You need to use the transpose of X, such that:

`X^T = ((a,c,e),(b,d,f))`

Performing the multiplication of matrices to the left, yields:

`((2,1),(1,4))*((a,c,e),(b,d,f))` = `((2a+b,2c+d,2e+f),(a+4b,c+4d,e+4f))`

Performing the multiplication of matrices to the right, yields:

`((0,0),(0,3))*((a,c,e),(b,d,f))` `= ((0,0,0),(3b,3d,3f))`

Adding the matrices to the right, yields:

`((-16,-4,0),(-8,-8,-20)) + ((0,0,0),(3b,3d,3f))` = `((-16,-4,0),(-8+3b,-8+3d,-20+3f))`

Equating matrices both sides yields:

`((2a+b,2c+d,2e+f),(a+4b,c+4d,e+4f)) = ((-16,-4,0),(-8+3b,-8+3d,-20+3f))`

Equating corresponding members yields:

`{(2a + b = -16),(2c + d = -4), (2e + f = 0),(a + 4b = -8 + 3b),(c + 4d = -8 + 3d),(e + 4f = -20 + 3f):}`

You need to solve the following systems of simultaneous equations, such that:

`{(2a + b = -16),(a + b = -8):} => a = -8 => b = 0`

`{(2c + d = -4),(c + d = -8):} => c = 4 => d = -12`

`{(2e + f = 0),(e + f = -20):} => e = 20 => f = -40`

**Hence, evaluating X, using it transpose `X^T` and performing matrices operations, yields **`X = ((-8,0),(4,-12),(20,-40)).`

Given matrix equation is

`[[2,1],[1,4]]X^T=-4[[4,1,0],[2,2,5]]+[[0,0],[0,3]]X^T`

or, `[[2,1],[1,4]]X^T-[[0,0],[0,3]]X^T=-4[[4,1,0],[2,2,5]]`

or, `{[[2,1],[1,4]]-[[0,0],[0,3]]}X^T=-4[[4,1,0],[2,2,5]]`

or, `[[2-0,1-0],[1-0,4-3]]X^T=-4[[4,1,0],[2,2,5]]`

or, `[[2,1],[1,1]]X^T=-4[[4,1,0],[2,2,5]]`

or, `X^T=-4[[2,1],[1,1]]^(-1)[[4,1,0],[2,2,5]]`

or, `X^T=-4[[1,-1],[-1,2]][[4,1,0],[2,2,5]]`

or, `X^T=-4[[2,-1,-5],[0,3,10]]`

or, `X^T=[[-4.2,-4.-1,-4.-5],[-4.0,-4.3,-4.10]]`

or, `X^T=[[-8,4,20],[0,-12,-40]]`

or, `X=[[-8,0],[4,-12],[20,-40]]` . Answer.