Linear Algebra Examples

$A [\begin{array}{c} 2 & 4 & 3 \\ 1 & 3 & 1 \\ 5 & 2 & 2 \\ 2 & 4 & 5 \\ 1 & 1 & 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 1

Multiply $A$ by each element of the matrix.

$[\begin{array}{c} A \cdot 2 & A \cdot 4 & A \cdot 3 \\ A \cdot 1 & A \cdot 3 & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2

Simplify each element in the matrix.

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Step 2.1

Move $2$ to the left of $A$ .

$[\begin{array}{c} 2 A & A \cdot 4 & A \cdot 3 \\ A \cdot 1 & A \cdot 3 & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.2

Move $4$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & A \cdot 3 \\ A \cdot 1 & A \cdot 3 & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.3

Move $3$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A \cdot 1 & A \cdot 3 & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.4

Multiply $A$ by $1$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & A \cdot 3 & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.5

Move $3$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \cdot 1 \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.6

Multiply $A$ by $1$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ A \cdot 5 & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.7

Move $5$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & A \cdot 2 & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.8

Move $2$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & A \cdot 2 \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.9

Move $2$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ A \cdot 2 & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.10

Move $2$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & A \cdot 4 & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.11

Move $4$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & A \cdot 5 \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.12

Move $5$ to the left of $A$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & 5 A \\ A \cdot 1 & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.13

Multiply $A$ by $1$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & 5 A \\ A & A \cdot 1 & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.14

Multiply $A$ by $1$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & 5 A \\ A & A & A \cdot 1 \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 2.15

Multiply $A$ by $1$ .

$[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & 5 A \\ A & A & A \end{array}] \cdot [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$

Step 3

Multiply $[\begin{array}{c} 2 A & 4 A & 3 A \\ A & 3 A & A \\ 5 A & 2 A & 2 A \\ 2 A & 4 A & 5 A \\ A & A & A \end{array}] [\begin{array}{c} 2 & 1 & 5 & 2 & 1 \\ 4 & 3 & 2 & 4 & 1 \\ 3 & 1 & 2 & 5 & 1 \end{array}]$ .

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Step 3.1

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 $5 \times 3$ and the second matrix is $3 \times 5$ .

Step 3.2

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

$[\begin{array}{c} 2 A \cdot 2 + 4 A \cdot 4 + 3 A \cdot 3 & 2 A \cdot 1 + 4 A \cdot 3 + 3 A \cdot 1 & 2 A \cdot 5 + 4 A \cdot 2 + 3 A \cdot 2 & 2 A \cdot 2 + 4 A \cdot 4 + 3 A \cdot 5 & 2 A \cdot 1 + 4 A \cdot 1 + 3 A \cdot 1 \\ A \cdot 2 + 3 A \cdot 4 + A \cdot 3 & A \cdot 1 + 3 A \cdot 3 + A \cdot 1 & A \cdot 5 + 3 A \cdot 2 + A \cdot 2 & A \cdot 2 + 3 A \cdot 4 + A \cdot 5 & A \cdot 1 + 3 A \cdot 1 + A \cdot 1 \\ 5 A \cdot 2 + 2 A \cdot 4 + 2 A \cdot 3 & 5 A \cdot 1 + 2 A \cdot 3 + 2 A \cdot 1 & 5 A \cdot 5 + 2 A \cdot 2 + 2 A \cdot 2 & 5 A \cdot 2 + 2 A \cdot 4 + 2 A \cdot 5 & 5 A \cdot 1 + 2 A \cdot 1 + 2 A \cdot 1 \\ 2 A \cdot 2 + 4 A \cdot 4 + 5 A \cdot 3 & 2 A \cdot 1 + 4 A \cdot 3 + 5 A \cdot 1 & 2 A \cdot 5 + 4 A \cdot 2 + 5 A \cdot 2 & 2 A \cdot 2 + 4 A \cdot 4 + 5 A \cdot 5 & 2 A \cdot 1 + 4 A \cdot 1 + 5 A \cdot 1 \\ A \cdot 2 + A \cdot 4 + A \cdot 3 & A \cdot 1 + A \cdot 3 + A \cdot 1 & A \cdot 5 + A \cdot 2 + A \cdot 2 & A \cdot 2 + A \cdot 4 + A \cdot 5 & A \cdot 1 + A \cdot 1 + A \cdot 1 \end{array}]$

Step 3.3

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

$[\begin{array}{c} 29 A & 17 A & 24 A & 35 A & 9 A \\ 17 A & 11 A & 13 A & 19 A & 5 A \\ 24 A & 13 A & 33 A & 28 A & 9 A \\ 35 A & 19 A & 28 A & 45 A & 11 A \\ 9 A & 5 A & 9 A & 11 A & 3 A \end{array}]$