Matrix Multiplication Flashcards

All 11 cards in this deck

What does it mean to say a matrix has order $2 \times 1$?

It has 2 rows and 1 column (a column matrix).

What are the steps to multiply a matrix by a scalar?

e.g. $3\begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix}$

1. Multiply every element by the scalar: $3\times1$, $3\times2$, $3\times0$, $3\times4$

2. Write the results in the same positions: $\begin{pmatrix} 3 & 6 \\ 0 & 12 \end{pmatrix}$

Write down the $2 \times 2$ identity matrix $I$.

$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Writing only "$I$" or "identity" is not accepted without this matrix.

What is the result of multiplying a $2 \times 2$ matrix $M$ by the identity matrix $I$?

$M$ itself, since $MI = IM = M$.

True or false? $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ is the identity matrix.

False. That is $-I$; the identity matrix is $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

What are the steps to multiply a $2 \times 2$ matrix by a $2 \times 1$ matrix?

e.g. $\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 4 \end{pmatrix}$

1. Top row times the column: $3 \times 1 + 5 \times 4 = 23$

2. Bottom row times the column: $1 \times 1 + 2 \times 4 = 9$

3. Write as a $2 \times 1$ matrix: $\begin{pmatrix} 23 \\ 9 \end{pmatrix}$

What are the steps to multiply a $2 \times 2$ matrix by a $2 \times 2$ matrix?

e.g. $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 4 & 0 \\ 1 & 5 \end{pmatrix}$

1. Each element = (row of 1st) $\times$ (column of 2nd), e.g. top left $= 1\times4 + 2\times1 = 6$

2. Repeat for the other three elements: $10$, $3$, $15$

3. Answer: $\begin{pmatrix} 6 & 10 \\ 3 & 15 \end{pmatrix}$

What are the steps to work out $M^2$?

e.g. $M = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$

1. Write $M^2 = M \times M$: $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$

2. Multiply row by column as usual: $\begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix}$

(For $M^3$, multiply $M^2$ by $M$.)

What are the steps to find unknown entries from a matrix multiplication?

e.g. $\begin{pmatrix} 3 & 5 \\ u & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} t \\ 6 \end{pmatrix}$

1. Form each row-by-column expression: $3\times1 + 5\times4 = t$ and $u\times1 + 2\times4 = 6$

2. Solve each equation: $t = 23$, $u = -2$

What are the steps to show a matrix product equals $kI$?

e.g. the product works out as $\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$

1. Multiply the matrices out fully.

2. Write the result as a multiple of the identity matrix: $-2\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

3. State $k$: $k = -2$ (the identity matrix must be shown).

True or false? For $2 \times 2$ matrices $A$ and $B$, $AB$ always equals $BA$.

False. Matrix multiplication is not commutative, so $AB$ and $BA$ are usually different.