Linear Algebra Examples

${[\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]}^{15}$

Step 1

To evaluate a square matrix to a positive integer power $n$ , multiply $n$ copies of the matrix.

$[\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 2

Multiply $[\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 2.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 2.2

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

$[\begin{array}{c} 10 \cdot 10 + 12 \cdot - 8 & 10 \cdot 12 + 12 \cdot - 10 \\ - 8 \cdot 10 - 10 \cdot - 8 & - 8 \cdot 12 - 10 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 2.3

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

$[\begin{array}{c} 4 & 0 \\ 0 & 4 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 3

Multiply $[\begin{array}{c} 4 & 0 \\ 0 & 4 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 3.1

Step 3.2

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

$[\begin{array}{c} 4 \cdot 10 + 0 \cdot - 8 & 4 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 4 \cdot - 8 & 0 \cdot 12 + 4 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 3.3

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

$[\begin{array}{c} 40 & 48 \\ - 32 & - 40 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 4

Multiply $[\begin{array}{c} 40 & 48 \\ - 32 & - 40 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 4.1

Step 4.2

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

$[\begin{array}{c} 40 \cdot 10 + 48 \cdot - 8 & 40 \cdot 12 + 48 \cdot - 10 \\ - 32 \cdot 10 - 40 \cdot - 8 & - 32 \cdot 12 - 40 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 4.3

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

$[\begin{array}{c} 16 & 0 \\ 0 & 16 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 5

Multiply $[\begin{array}{c} 16 & 0 \\ 0 & 16 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 5.1

Step 5.2

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

$[\begin{array}{c} 16 \cdot 10 + 0 \cdot - 8 & 16 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 16 \cdot - 8 & 0 \cdot 12 + 16 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 5.3

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

$[\begin{array}{c} 160 & 192 \\ - 128 & - 160 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 6

Multiply $[\begin{array}{c} 160 & 192 \\ - 128 & - 160 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 6.1

Step 6.2

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

$[\begin{array}{c} 160 \cdot 10 + 192 \cdot - 8 & 160 \cdot 12 + 192 \cdot - 10 \\ - 128 \cdot 10 - 160 \cdot - 8 & - 128 \cdot 12 - 160 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 6.3

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

$[\begin{array}{c} 64 & 0 \\ 0 & 64 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 7

Multiply $[\begin{array}{c} 64 & 0 \\ 0 & 64 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 7.1

Step 7.2

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

$[\begin{array}{c} 64 \cdot 10 + 0 \cdot - 8 & 64 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 64 \cdot - 8 & 0 \cdot 12 + 64 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 7.3

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

$[\begin{array}{c} 640 & 768 \\ - 512 & - 640 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 8

Multiply $[\begin{array}{c} 640 & 768 \\ - 512 & - 640 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 8.1

Step 8.2

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

$[\begin{array}{c} 640 \cdot 10 + 768 \cdot - 8 & 640 \cdot 12 + 768 \cdot - 10 \\ - 512 \cdot 10 - 640 \cdot - 8 & - 512 \cdot 12 - 640 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 8.3

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

$[\begin{array}{c} 256 & 0 \\ 0 & 256 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 9

Multiply $[\begin{array}{c} 256 & 0 \\ 0 & 256 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 9.1

Step 9.2

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

$[\begin{array}{c} 256 \cdot 10 + 0 \cdot - 8 & 256 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 256 \cdot - 8 & 0 \cdot 12 + 256 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 9.3

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

$[\begin{array}{c} 2560 & 3072 \\ - 2048 & - 2560 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 10

Multiply $[\begin{array}{c} 2560 & 3072 \\ - 2048 & - 2560 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 10.1

Step 10.2

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

$[\begin{array}{c} 2560 \cdot 10 + 3072 \cdot - 8 & 2560 \cdot 12 + 3072 \cdot - 10 \\ - 2048 \cdot 10 - 2560 \cdot - 8 & - 2048 \cdot 12 - 2560 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 10.3

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

$[\begin{array}{c} 1024 & 0 \\ 0 & 1024 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 11

Multiply $[\begin{array}{c} 1024 & 0 \\ 0 & 1024 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 11.1

Step 11.2

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

$[\begin{array}{c} 1024 \cdot 10 + 0 \cdot - 8 & 1024 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 1024 \cdot - 8 & 0 \cdot 12 + 1024 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 11.3

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

$[\begin{array}{c} 10240 & 12288 \\ - 8192 & - 10240 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 12

Multiply $[\begin{array}{c} 10240 & 12288 \\ - 8192 & - 10240 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 12.1

Step 12.2

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

$[\begin{array}{c} 10240 \cdot 10 + 12288 \cdot - 8 & 10240 \cdot 12 + 12288 \cdot - 10 \\ - 8192 \cdot 10 - 10240 \cdot - 8 & - 8192 \cdot 12 - 10240 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 12.3

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

$[\begin{array}{c} 4096 & 0 \\ 0 & 4096 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 13

Multiply $[\begin{array}{c} 4096 & 0 \\ 0 & 4096 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 13.1

Step 13.2

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

$[\begin{array}{c} 4096 \cdot 10 + 0 \cdot - 8 & 4096 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 4096 \cdot - 8 & 0 \cdot 12 + 4096 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 13.3

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

$[\begin{array}{c} 40960 & 49152 \\ - 32768 & - 40960 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 14

Multiply $[\begin{array}{c} 40960 & 49152 \\ - 32768 & - 40960 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 14.1

Step 14.2

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

$[\begin{array}{c} 40960 \cdot 10 + 49152 \cdot - 8 & 40960 \cdot 12 + 49152 \cdot - 10 \\ - 32768 \cdot 10 - 40960 \cdot - 8 & - 32768 \cdot 12 - 40960 \cdot - 10 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 14.3

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

$[\begin{array}{c} 16384 & 0 \\ 0 & 16384 \end{array}] \cdot [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$

Step 15

Multiply $[\begin{array}{c} 16384 & 0 \\ 0 & 16384 \end{array}] [\begin{array}{c} 10 & 12 \\ - 8 & - 10 \end{array}]$ .

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Step 15.1

Step 15.2

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

$[\begin{array}{c} 16384 \cdot 10 + 0 \cdot - 8 & 16384 \cdot 12 + 0 \cdot - 10 \\ 0 \cdot 10 + 16384 \cdot - 8 & 0 \cdot 12 + 16384 \cdot - 10 \end{array}]$

Step 15.3

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

$[\begin{array}{c} 163840 & 196608 \\ - 131072 & - 163840 \end{array}]$

Step 16

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$163840 \cdot - 163840 - (- 131072 \cdot 196608)$

Step 17

Simplify the determinant.

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Step 17.1

Simplify each term.

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Step 17.1.1

Multiply $163840$ by $- 163840$ .

$- 26843545600 - (- 131072 \cdot 196608)$

Step 17.1.2

Multiply $- (- 131072 \cdot 196608)$ .

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Step 17.1.2.1

Multiply $- 131072$ by $196608$ .

$- 26843545600 - - 25769803776$

Step 17.1.2.2

Multiply $- 1$ by $- 25769803776$ .

$- 26843545600 + 25769803776$

Step 17.2

Add $- 26843545600$ and $25769803776$ .

$- 1073741824$