Linear Algebra Examples

$C [\begin{array}{c} 1 & 0 \\ 4 & 2 \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $C$ by each element of the matrix.

$[\begin{array}{c} C \cdot 1 & C \cdot 0 \\ C \cdot 4 & C \cdot 2 \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $C$ by $1$ .

$[\begin{array}{c} C & C \cdot 0 \\ C \cdot 4 & C \cdot 2 \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1.2.2

Multiply $C$ by $0$ .

$[\begin{array}{c} C & 0 \\ C \cdot 4 & C \cdot 2 \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1.2.3

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

$[\begin{array}{c} C & 0 \\ 4 C & C \cdot 2 \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1.2.4

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

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + D [\begin{array}{c} 4 & 5 \\ 2 & 3 \end{array}]$

Step 1.3

Multiply $D$ by each element of the matrix.

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + [\begin{array}{c} D \cdot 4 & D \cdot 5 \\ D \cdot 2 & D \cdot 3 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

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

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + [\begin{array}{c} 4 D & D \cdot 5 \\ D \cdot 2 & D \cdot 3 \end{array}]$

Step 1.4.2

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

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + [\begin{array}{c} 4 D & 5 D \\ D \cdot 2 & D \cdot 3 \end{array}]$

Step 1.4.3

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

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + [\begin{array}{c} 4 D & 5 D \\ 2 D & D \cdot 3 \end{array}]$

Step 1.4.4

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

$[\begin{array}{c} C & 0 \\ 4 C & 2 C \end{array}] + [\begin{array}{c} 4 D & 5 D \\ 2 D & 3 D \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} C + 4 D & 0 + 5 D \\ 4 C + 2 D & 2 C + 3 D \end{array}]$

Step 3

Add $0$ and $5 D$ .

$[\begin{array}{c} C + 4 D & 5 D \\ 4 C + 2 D & 2 C + 3 D \end{array}]$