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iris [78.8K]
2 years ago
15

How to solve the matrix equation for matrix Y.simplify all elements

Mathematics
1 answer:
marta [7]2 years ago
4 0

Given <em>YA</em> = <em>B</em>, you can solve for <em>Y</em> by multiplying by <em>A </em>⁻¹ on the right (on both sides of the equation). So we have

<em>YA</em> = <em>B</em>   ==>   (<em>YA</em>) <em>A </em>⁻¹ = <em>BA </em>⁻¹   ==>   <em>Y</em> (<em>AA </em>⁻¹) = <em>BA </em>⁻¹   ==>   <em>Y</em> = <em>BA </em>⁻¹

provided that the inverse of <em>A</em> exists. In this case, det(<em>A</em>) = 5 ≠ 0, so the inverse does exist, and

A=\begin{bmatrix}-1&-4\\0&-5\end{bmatrix} \implies A^{-1}=\dfrac1{\det(A)}\begin{bmatrix}-5&0\\4&-1\end{bmatrix} = \begin{bmatrix}-1&0\\\frac45&-\frac15\end{bmatrix}

Then

Y=\begin{bmatrix}5&-5\\8&-8\end{bmatrix}A^{-1} = \begin{bmatrix}-5&5\\-8&\frac{24}5\end{bmatrix}

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