Linear Algebra Examples

3

$3$

Step 1

The identity matrix or unit matrix of size

3

$3$ is the

3 \times 3

$3 \times 3$ square matrix with ones on the main diagonal and zeros elsewhere.

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 2

Multiply the scalar

3

$3$ by the identity matrix.

3 ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

$3 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 3

Simplify.

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Step 3.1

Multiply

3

$3$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 3 \cdot 1 & 3 \cdot 0 & 3 \cdot 0 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 3 \cdot 1 & 3 \cdot 0 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2

Simplify each element in the matrix.

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Step 3.2.1

Multiply

3

$3$ by

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} 3 & 3 \cdot 0 & 3 \cdot 0 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 3 & 3 \cdot 0 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.2

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.3

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.4

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.5

Multiply

$3$ by

$1$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 & 3 \cdot 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.6

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 3 \cdot 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.7

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 3 \cdot 0 & 3 \cdot 1 \end{array}]$

Step 3.2.8

Multiply

$3$ by

$0$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \cdot 1 \end{array}]$

Step 3.2.9

Multiply

$3$ by

$1$ .

$[\begin{array}{c} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}]$