Study guide and
13 practice problems
on:
Matrix multiplication
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The matrix product of an $n \times m$ matrix with an $m \times \ell$ matrix is an $n \times \ell$ matrix.
The $(i,j)$ entry of the matrix product $\bfA \mathbf{B}$ is the dot product of the $i$th row of $\bfA$ with the $j$th column of $\mathbf{B}$.
The $(i,j)$ entry of the matrix product $AB$ is $(AB)_{ij} = \sum_k A_{ik} B_{kj}.$
Related topics
Matrices and linear equations
(20 problems)
Multivariable calculus
(147 problems)
Practice problems
Show that matrix multiplication is associative. That is, show that $(AB)C = A(BC)$ for any matrices $A$, $B$, and $C$ that are of the appropriate dimensions for matrix multiplication.
Solution
Compute the matrix multiplications $$\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\begin{pmatrix} 1 \\2\\3\end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 \\2\\3\end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}.$$
Solution
Compute the matrix multiplication $$ \begin{pmatrix}1 & 0 & 2 \\ -1 & 1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$$
Solution
Find the $3 \times 3$ matrix $\bfA$ satisfying \begin{align}
\bfA \begin{pmatrix}1\\0\\0\end{pmatrix} &= \begin{pmatrix}1\\2\\3\end{pmatrix}, \quad
\bfA \begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}4\\5\\6\end{pmatrix}, \quad
\bfA \begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}7\\8\\9\end{pmatrix}.\\
\end{align}
Solution
An orthogonal matrix is one satisfying $A A^t = I$. Suppose $$A = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ a & b & c \end{pmatrix}.$$
If $A$ is orthogonal, show that $(a, b, c)$ is perpendicular to $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$ and $(0,\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$
If $A$ is orthogonal, show that $(a,b,c)$ is of unit length.
Find two values of $(a, b, c)$ so that $A$ is orthogonal.
Solution
Write this matrix equation as a system of 3 equations. Solve for $x,y,z$: $$\begin{pmatrix}1 & 1& 1\\0 & 1 & 1\\ 0 &0 & 1 \end{pmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}4\\3\\1\end{pmatrix}$$
Solution
Find the second degree polynomial going through $(-1, 1), (1, 3),$ and $(2,2)$.
Hint: To find the coefficients of $y = a + bx + cx^2$, set up a $3 \times 3$ matrix satisfied by $a,b,c$.
Solution
Write the following system as a matrix equation for $x,y,z$:\begin{align}
y +z &=4\\
2x -y &=z
\end{align}
Solution
Solve by matrix inversion: $$\begin{pmatrix} 2 & 3 \\ 10 & 16 \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}.$$
Solution
Let $A= \begin{pmatrix}1/2 & 0 \\ 0 & 2 \end{pmatrix}$.
How does the following shape get transformed by application of $A$:
Solution
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