Solve the Matrix Equation [[1,2,0],[2,5,7],[3,-3,-1]]*[[2,0,5],[-2,2,0],[3,1,5]]=[[a_11,4,5],[15,a_22,45],[9,-7,a_33]]

$[\begin{array}{c} 1 & 2 & 0 \\ 2 & 5 & 7 \\ 3 & - 3 & - 1 \end{array}] \cdot [\begin{array}{c} 2 & 0 & 5 \\ - 2 & 2 & 0 \\ 3 & 1 & 5 \end{array}] = [\begin{array}{c} a_{11} & 4 & 5 \\ 15 & a_{22} & 45 \\ 9 & - 7 & a_{33} \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 2 & 0 \\ 2 & 5 & 7 \\ 3 & - 3 & - 1 \end{array}] [\begin{array}{c} 2 & 0 & 5 \\ - 2 & 2 & 0 \\ 3 & 1 & 5 \end{array}]$ .

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 \cdot 2 + 2 \cdot - 2 + 0 \cdot 3 & 1 \cdot 0 + 2 \cdot 2 + 0 \cdot 1 & 1 \cdot 5 + 2 \cdot 0 + 0 \cdot 5 \\ 2 \cdot 2 + 5 \cdot - 2 + 7 \cdot 3 & 2 \cdot 0 + 5 \cdot 2 + 7 \cdot 1 & 2 \cdot 5 + 5 \cdot 0 + 7 \cdot 5 \\ 3 \cdot 2 - 3 \cdot - 2 - 1 \cdot 3 & 3 \cdot 0 - 3 \cdot 2 - 1 \cdot 1 & 3 \cdot 5 - 3 \cdot 0 - 1 \cdot 5 \end{array}] = [\begin{array}{c} a_{11} & 4 & 5 \\ 15 & a_{22} & 45 \\ 9 & - 7 & a_{33} \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} - 2 & 4 & 5 \\ 15 & 17 & 45 \\ 9 & - 7 & 10 \end{array}] = [\begin{array}{c} a_{11} & 4 & 5 \\ 15 & a_{22} & 45 \\ 9 & - 7 & a_{33} \end{array}]$

Step 2

Write as a linear system of equations.

$- 2 = a_{11} 4 = 4 5 = 5 15 = 15 17 = a_{22} 45 = 45 9 = 9 - 7 = - 7 10 = a_{33}$

Step 3

Since $- 7 = - 7$ is always true, the matrix equation has infinite solutions.

Infinite solutions satisfying the system.

Solve the Matrix Equation [[1,2,0],[2,5,7],[3,-3,-1]]*[[2,0,5],[-2,2,0],[3,1,5]]=[[a_11,4,5],[15,a_22,45],[9,-7,a_33]]

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