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Section 3.5 Matrices

From now on assume all linear spaces are finite dimensional and a basis, \(\beta\text{,}\) has been chosen. Each point \(\mathbf{P}\) in the linear space can then be uniquely identified using the list of coefficients used to write \(\mathbf{P}\) as a linear combination of the points in \(\beta\text{.}\) This list is called its coordinate vector and is written

\begin{equation*} _{\beta} = \left[ \begin{array}{c} c_1\\c_2\\ \vdots \\c_n \end{array} \right]\in\mathbb{R}^n \end{equation*}

Example. If \(\beta=\{1,x,x^2\}\) and \(\alpha = \{1+x+x^2,x+x^2,x^2\}\)

\begin{equation*} [5+4x+3x^2]_{\beta}= \left[ \begin{array}{c} 5\\4\\3 \end{array} \right] \ \ \text{ and } \ \ [5+4x+3x^2]_{\alpha}= \left[ \begin{array}{r} 5\\-1\\-1 \end{array} \right] \end{equation*}

Lists of coordinate vectors can be organized into what are called coordinate matrices.

Definition 3.5.2.

A Matrix is a rectangular array of numbers, written

\begin{equation*} \mtx{A}_{r \times c}= [a_{ij}] =\left[ \begin{array}{cccc} a_{11} \amp a_{12} \amp \cdots \amp a_{1c} \\ a_{21} \amp a_{22} \amp \cdots \amp a_{2c} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ a_{r1} \amp a_{r2} \amp \cdots \amp a_{rc} \end{array} \right] \end{equation*}
Definition 3.5.3.

For matrices \(\mtx{A}_{r \times c} =[a_{ij}]\) and \(\mtx{B}_{ m \times n} = [b_{ij}]\)

  1. The sum of \(\mtx{A}\) and \(\mtx{B}\) is possible when \(r=m\) and \(c=n\) and is defined by

    \begin{equation*} \mtx{A}+\mtx{B} = [s_{ij}] \text{ where } s_{ij}=a_{ij} + b_{ij} \end{equation*}

  2. The scalar product of \(t \in \mathbb{R}\) and \(\mtx{A}\) is defined by

    \begin{equation*} t\mtx{A} = [q_{ij}] \text{ where } q_{ij} = ta_{ij} \end{equation*}

  3. The product of \(\mtx{A}\) and \(\mtx{B}\) is possible when \(c=m\) and is defined by

    \begin{equation*} \mtx{AB} = [p_{ij}] \text{ where } p_{ij} = a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj} \end{equation*}

Perform, if possible, the indicated matrix operations.

\begin{equation*} \begin{array}{ccc} A=\left[ \begin{array}{ccc} 1\amp 2\amp 3 \\ 4\amp 5\amp 6\\ 7\amp 8\amp 9 \end{array} \right] \amp B=\left[ \begin{array}{rrr} -1\amp 0\amp 1 \\ 1\amp -1\amp 0\\ 0\amp 0\amp 0 \end{array} \right] \amp C=\left[ \begin{array}{ccc} 1\amp 10\amp 100 \\ 0\amp -1\amp 0 \end{array} \right] \end{array} \end{equation*}

  1. \(\displaystyle 3A=\)

  2. \(\displaystyle 3+A =\)

  3. \(\displaystyle A+B =\)

  4. \(\displaystyle A+C =\)

  5. \(\displaystyle AB =\)

  6. \(\displaystyle BA =\)

  7. \(\displaystyle AC =\)

  8. \(\displaystyle CA =\)

Definition 3.5.5.

An additive identity for the set of \(n \times n\) matrices is an \(n \times n\) matrix \(\mtx{O}\) such that for all \(n \times n\) matrices \(\mtx{M}\text{,}\) \(\mtx{M}+\mtx{O} = \mtx{M}\text{.}\) \(\mtx{O}\) is also called the zero matrix. A multiplicative identity for the set of \(n \times n\) matrices is an \(n \times n\) matrix \(\mtx{I}\) such that for all \(n \times n\) matrices \(\mtx{M}\text{,}\) \(\mtx{MI} = \mtx{M}\) and \(\mtx{IM} = \mtx{M}\text{.}\)

Show that for the set of \(3 \times 3\) matrices there exists a unique additive identity and a unique multiplicative identity.

Definition 3.5.7.

A multiplicative inverse of a matrix \(\mtx{A}\) is a matrix \(\mtx{A}^{-1}\) such that \(\mtx{AA}^{-1}=\mtx{I}\) and \(\mtx{A}^{-1}\mtx{A}=\mtx{I}\) .

Does a multiplicative inverse exist for all matrices? Is the multiplicative inverse of a given matrix unique?

Where else is the notation \((\ )^{-1}\) used? Is this notation used consistently?