Let $A = \begin{bmatrix} 1 & 4 & -3 \\ 6 & 3 & 0\end{bmatrix}$, $B = \begin{bmatrix} 3 & 2 & 1 \\ -2 & -6 & 5\end{bmatrix}$ and $C = \begin{bmatrix} 2 & 4 \\ 4 & 0 \\ -2 & 2\end{bmatrix}$. Let $\alpha = 4$ and $\beta = 1/2$. Perform the following calculations: (1) $A + B$, (2) $A + C$, (3) $\transpose{B} + C$, (4) $A + \transpose{B}$, (5) $\beta C$, (6) $4A - 3B$, (7) $\transpose{A} + \alpha C$, (8) $A + B - \transpose{C}$, (9) $4A + 2B - 5\transpose{C}$.
Solve the given vector equation for $x$, or explain why no solution exists: \begin{equation*} 2 \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 2 \end{bmatrix} - 3 \begin{bmatrix} 1 & 1 & 2 \\ 0 & 1 & x \end{bmatrix} = \begin{bmatrix} -1 & 1 & 0 \\ 0 & 5 & -2 \end{bmatrix} \end{equation*}
Solve the given matrix equation for $\alpha$, or explain why no solution exists: \begin{equation*} \alpha\begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & -1 \end{bmatrix} + \begin{bmatrix} 4 & 3 & -6 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 12 & 6 \\ 6 & 4 & -2 \end{bmatrix} \end{equation*}
Solve the given matrix equation for $\alpha$, or explain why no solution exists: \begin{equation*} \alpha \begin{bmatrix} 3 & 1 \\ 2 & 0 \\ 1 & 4\end{bmatrix} - \begin{bmatrix} 4 & 1 \\ 3 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & -2 \\ 2 & 6 \end{bmatrix} \end{equation*}
Find $\alpha$ and $\beta$ that solve the following equation: \begin{equation*} \alpha\begin{bmatrix} 1 & 2 \\ 4 & 1 \end{bmatrix} + \beta\begin{bmatrix} 2 & 1 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 4 \\ 6 & 1 \end{bmatrix} \end{equation*}
In Chapter V we defined the operations of vector addition and vector scalar multiplication in Definition CVA and Definition CVSM. These two operations formed the underpinnings of the remainder of the chapter. We have now defined similar operations for matrices in Definition MA and Definition MSM. You will have noticed the resulting similarities between Theorem VSPCV and Theorem VSPM.
In Exercises M20–M25, you will be asked to extend these similarities to other fundamental definitions and concepts we first saw in Chapter V. This sequence of problems was suggested by Martin Jackson.
Suppose $S=\set{B_1,\,B_2,\,B_3,\,\ldots,\,B_p}$ is a set of matrices from $M_{mn}$. Formulate appropriate definitions for the following terms and give an example of the use of each.
Show that the set $S$ is linearly independent in $M_{22}$. \begin{equation*} S=\set{ \begin{bmatrix}1&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&1\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&1\end{bmatrix} } \end{equation*}
Determine if the set $S$ below is linearly independent in $M_{23}$. \begin{equation*} \set{ \begin{bmatrix} -2 & 3 & 4 \\ -1 & 3 & -2 \end{bmatrix},\, \begin{bmatrix} 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix},\, \begin{bmatrix}-1 & -2 & -2 \\ 2 & 2 & 2 \end{bmatrix},\, \begin{bmatrix}-1 & 1 & 0 \\ -1 & 0 & -2 \end{bmatrix},\, \begin{bmatrix}-1 & 2 & -2 \\ 0 & -1 & -2 \end{bmatrix} } \end{equation*}
Determine if the matrix $A$ is in the span of $S$. In other words, is $A\in\spn{S}$? If so write $A$ as a linear combination of the elements of $S$. \begin{align*} A&= \begin{bmatrix} -13 & 24 & 2\\ -8 & -2 & -20 \end{bmatrix}\\ S&=\set{ \begin{bmatrix} -2 & 3 & 4 \\ -1 & 3 & -2 \end{bmatrix},\, \begin{bmatrix} 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix},\, \begin{bmatrix}-1 & -2 & -2 \\ 2 & 2 & 2 \end{bmatrix},\, \begin{bmatrix}-1 & 1 & 0 \\ -1 & 0 & -2 \end{bmatrix},\, \begin{bmatrix}-1 & 2 & -2 \\ 0 & -1 & -2 \end{bmatrix} } \end{align*}
Suppose $Y$ is the set of all $3\times 3$ symmetric matrices (Definition SYM). Find a set $T$ so that $T$ is linearly independent and $\spn{T}=Y$.
Define a subset of $M_{33}$ by \begin{equation*} U_{33}=\setparts{ A\in M_{33} }{ \matrixentry{A}{ij}=0\text{ whenever }i>j } \end{equation*} Find a set $R$ so that $R$ is linearly independent and $\spn{R}=U_{33}$.
Prove Property CM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.
Prove Property AAM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.
Prove Property SMAM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.
Prove Property DMAM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.
A matrix $A$ is skew-symmetric if $\transpose{A}=-A$ Exercises T30–T37 employ this definition.
Prove that a skew-symmetric matrix is square. (Hint: study the proof of Theorem SMS.)
Prove that a skew-symmetric matrix must have zeros for its diagonal elements. In other words, if $A$ is skew-symmetric of size $n$, then $\matrixentry{A}{ii}=0$ for $1\leq i\leq n$. (Hint: carefully construct an example of a $3\times 3$ skew-symmetric matrix before attempting a proof.)
Prove that a matrix $A$ is both skew-symmetric and symmetric if and only if $A$ is the zero matrix. (Hint: one half of this proof is very easy, the other half takes a little more work.)
Suppose $A$ and $B$ are both skew-symmetric matrices of the same size and $\alpha,\,\beta\in\complexes$. Prove that $\alpha A + \beta B$ is a skew-symmetric matrix.
Suppose $A$ is a square matrix. Prove that $A+\transpose{A}$ is a symmetric matrix.
Suppose $A$ is a square matrix. Prove that $A-\transpose{A}$ is a skew-symmetric matrix.
Suppose $A$ is a square matrix. Prove that there is a symmetric matrix $B$ and a skew-symmetric matrix $C$ such that $A=B+C$. In other words, any square matrix can be decomposed into a symmetric matrix and a skew-symmetric matrix (Proof Technique DC). (Hint: consider building a proof on Exercise MO.T34 and Exercise MO.T35.)
Prove that the decomposition in Exercise MO.T36 is unique (see Proof Technique U). (Hint: a proof can turn on Exercise MO.T31.)