Section 2.1 Matrix Addition, Scalar Multiplication, and Transposition

Definition: 1. For any
$m\times n$ matrix, $A=\left[\begin{array}{c}\overrightarrow{v_1}\cdots \overrightarrow{v_n}\end{array}\right]$ the i-th entry of $\overrightarrow{v_j}$ vector is called the (i, j)-entry of

$\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \cdots & a_{1j}& \cdots & a_{1n}\\ a_{21}& & & & & a_{2n}\\ ⋮& & & & & ⋮\\ a_{i1}& a_{i2}& \cdots & a_{ij}& \cdots & a_{in}\\ ⋮& & & & & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mj}& \cdots & a_{mn}\end{array}\right]$

2. A square matrix is an $n\times n$ matrix.

3. The diagonal entries in an $m\times n$ matrix $A=\left[\begin{array}{c}\overrightarrow{a_{ij}}\end{array}\right]$ are $a_{11},a_{22},\cdots$, and they form the main diagonal of $A$. A diagonal matrix is a square $n\times n$ matrix whose non-diagonal entries are zero

4. The matrix with 1′s on the diagonal and 0′s elsewhere is called an identity matrix and is denoted by I.

5. A zero matrix is a $m\times n$ matrix whose entries are all zero and is written as 0.

6. $A=\left[\begin{array}{c}\overrightarrow{a_{ij}}\end{array}\right]$ and $B=\left[\begin{array}{c}\overrightarrow{b_{ij}}\end{array}\right]$ are two $m\times n$ matrices. We say $A$ is equal to $B$ if
$a_{ij}=b_{ij}$ for all $i,j$. The sum of $A$ and $B$ is
$A+B=\left[\begin{array}{c}\overrightarrow{a_{ij}}+\overrightarrow{b_{ij}}\end{array}\right]$.

7. If $r$ is a scalar and $A=\left[\begin{array}{c}\overrightarrow{a_{ij}}\end{array}\right]$
is a matrix, then the scalar multiple $rA=\left[\begin{array}{c}\overrightarrow{r{a}_{ij}}\end{array}\right]$ which entries are $r$ times the entries of $A$.

Note: Only when two matrices of the same size can they be equal. The sum of two matrices is only defined when two matrices are of the same size.

Theorem: $A,B,$ and $C$ are matrices of the same size, and let $r$ and $s$ be scalars.

(a) $A+B=B+A$

(b) $(A+B)+C=A+(B+C)$

(c) $A+0=A$

(d) $r(A+B)=rA+rB$

(e) $(r+d)A=rA+sA$

(f) $r(sA)=(rs)A$

Definition: 1. The transpose of $A,A^T$ is the matrix that has rows of $A$ as its columns or has columns of $A$ as its rows.

2. The matrix $A$ is called symmetric if and only if $A=A^T$. Note that this immediately implies that $A$ is a square matrix.

Theorem: Let $A$ and $B$ be matrices whose sizes are appropiate for the sums of products. Then

a. $(A^T{)}^T=A$

b. $(A+B{)}^T=A^T+B^T$

c. For any scalar, $r,(rA{)}^T=r{A}^T$

Example 1: Verify $(A+B{)}^T=A^T+B^T$.

Exercise 1: Verify $((A{)}^T{)}^T=A$.

Example 2: Find $(2A+B^T{)}^T$ where $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$.

Exercise 2: Find $(A^T-3B{)}^T$ where $A=\left[\begin{array}{cc}1& -2\\ -2& 0\\ 3& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 3& 4\\ -2& 1& 1\end{array}\right]$.

Group Work 1: Show $A+2B$ is a diagonal matrix if both $A$ and $B$ are diagonal matrices.

Group Work 2: In each case either show that the statement is true or give an
example showing it is false.

a. If $A+B=A+C$, then $B$ and $C$ have the same size.

b. If $A+B=0$, then $B=0$.

c. If the (3,1)-entry of $A$ is 5, then the (1,3)-entry of $A^T$ is 5.

d. $A$ and $A^T$ have the same main diagonal for every matrix $A$.

e. If $B$ is symmetric and $A^T=3B$, then $A=3B$.

f. If $A$ and $B$ are symmetric, then $kA+mB$ is symmetric for any scalars $k$ and $m$.

g. $A+A^T$ is symmetric for any square matrix $A$.

h. If $Q+A=A$ holds for every $m\times n$ matrix $A$, then $Q$
is the zero matrix.

Group Work 3: Show $A^T+3B$ is a symmetric matrix if both $A$ and $B$ are symmetric matrices