How to multiply a matrix by a scalar and use zero matrices

A scalar changes every entry of a matrix by the same factor. This lets us scale a whole array while keeping its dimensions. We’ll use scalar multiplication in a matrix equation and then connect it with zero matrices.

Distribute the scalar to every entry

A = [\begin{array}{r} 6 & 2 \\ - 1 & - 4 \end{array}]

3 A = [\begin{array}{r} 18 & 6 \\ - 3 & - 12 \end{array}]

Multiplying by $1 / 6$ divides each entry by 6.

\frac{1}{6} A = [\begin{array}{r} 1 & \frac{1}{3} \\ - \frac{1}{6} & - \frac{2}{3} \end{array}]

Solve a matrix equation after scaling

Solve $2 P + K = - 3 Q$ , where

P = [\begin{array}{r} 5 & - 7 \\ - 1 & 0 \end{array}]

Q = [\begin{array}{r} 2 & 4 \\ 11 & - 9 \end{array}]

The scaled matrices have rows $(10, - 14)$ , $(- 2, 0)$ for $2 P$ , and $(- 6, - 12)$ , $(- 33, 27)$ for $- 3 Q$ . Subtract $2 P$ from both sides to get $K = - 3 Q - 2 P$ .

K = [\begin{array}{r} - 16 & 2 \\ - 31 & 27 \end{array}]

Recognize a zero matrix

A zero matrix has zero in every position. The notation $O_{2 \times 3}$ specifies two rows and three columns.

O_{2 \times 3} = [\begin{array}{r} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]

For matching dimensions, $A + O = A$ and $A - O = A$ . Multiplying any matrix by the scalar zero gives a zero matrix of the same dimensions.

Add opposite matrices

The opposite matrix $- A$ is found by multiplying every entry of $A$ by $- 1$ . Corresponding entries cancel in $A + (- A)$ , giving the zero matrix. Also, $O - A = - A$ , since subtraction retains the sign of the matrix being subtracted.

Example: Add a matrix and its opposite

Use this two-by-three matrix.

K = [\begin{array}{r} 9 & - 6 & 2 \\ 1 & 0 & - 7 \end{array}]

To find $- K$ , change the sign of every entry in $K$ .

- K = [\begin{array}{r} - 9 & 6 & - 2 \\ - 1 & 0 & 7 \end{array}]

When we add $K + (- K)$ , each entry cancels with its opposite. For instance, the second entry in the first row is $- 6 + 6 = 0$ . The result is $O_{2 \times 3}$ .

Learn mathKrista KingAugust 19, 2021math, learn online, online course, online math, equilibrium solutions