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1.C Inverse matrices and triangular matrices

Recall that the inverse of a square matrix $A\in\operatorname{M}_{{n}}({F})$ , is another square matrix $B\in\operatorname{M}_{{n}}({F})$ such that $AB=BA=I_{n}$ , where $I_{n}$ is the identity matrix. Here $F$ is a field, such as $F={\mathbb{R}}$ . One can prove that if an inverse matrix exists then it is unique, so we are allowed to call it the inverse (see also Exercise 1.22), and we usually denote it by $A^{-1}$ . If $A$ has an inverse, then we say it is invertible.

Theorem 1.12.

Let $F$ be a field. If $A\in\operatorname{M}_{{n}}({F})$ , then $A$ is invertible if and only if $\det(A)\neq 0$ .

We will not recall the definition of the determinant here. For that, see the MATH105 course notes.

When $F={\mathbb{R}}$ , the above theorem should be familiar. But consider what it says when $F={\mathbb{Q}}$ . It says that the inverse matrix always has coefficients in the rational numbers, if your original matrix had only coefficients in the rational numbers. If you consider the steps involved during the algorithm for inverting matrices, this should come as no surprise.

Example 1.13.

Find the inverse of the matrix $A=\begin{bmatrix}3&0&-1\\ 1&-1&1\\ 0&0&-1\end{bmatrix}\in\operatorname{M}_{{3}}({{\mathbb{Q}}})$ .

Solution: We make the augmented matrix $[A|I_{3}]$ , and find its reduced row echelon form, using e.r.o.’s (details not shown).

\left[\begin{array}[]{ccc|ccc}3&0&-1&1&0&0\\ 1&-1&1&0&1&0\\ 0&0&-1&0&0&1\end{array}\right]\rightarrow\cdots\rightarrow\left[\begin{array}[% ]{ccc|ccc}1&0&0&\frac{1}{3}&0&-\frac{1}{3}\\ 0&1&0&\frac{1}{3}&-1&-\frac{4}{3}\\ 0&0&1&0&0&-1\end{array}\right].

Then the $3\times 3$ matrix on the far right is the inverse of $A$ . Since it is easy to make a mistake, it is always worth checking:

\begin{bmatrix}3&0&-1\\ 1&-1&1\\ 0&0&-1\end{bmatrix}\begin{bmatrix}\frac{1}{3}&0&-\frac{1}{3}\\ \frac{1}{3}&-1&-\frac{4}{3}\\ 0&0&-1\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}.

Notice that $A$ had all coefficients in $F={\mathbb{Q}}$ , and therefore $A^{-1}$ must also have all coefficients in $F={\mathbb{Q}}$ (and it does).

Exercise 1.14:

For the field $\mathbb{F}_{5}=\{0,1,2,3,4\}$ of integers modulo 5, from Exercise 1.2, find the inverse of the matrix $A:=\begin{bmatrix}1&2\\ 3&4\end{bmatrix}\in\operatorname{M}_{{2}}({\mathbb{F}_{5}})$ . Note your answer must be a matrix whose entries are in the field $\mathbb{F}_{5}$ . Check your answer is correct by multiplying $A\cdot A^{-1}=I_{2}$ .

[End of Exercise]

Recall from MATH105 that for a matrix $A=[a_{ij}]$ , the entries $a_{11},a_{22},a_{33},\cdots,a_{nn}$ are collectively called the diagonal of the matrix. Also, $A$ is called a diagonal matrix if $a_{ij}=0$ whenever $i\neq j$ . The matrix is called upper triangular (respectively lower triangular) if the only non-zero entries occur on or above (resp. below) the diagonal. We will use the same terminology for matrices with coefficients in any field.

Exercise 1.15:

Consider the following matrices in $\operatorname{M}_{{3}}({F})$ :

$\begin{bmatrix}1&5&10\\ 2&3&-5\\ 4&5&6\end{bmatrix}$
$\begin{bmatrix}2&0&0\\ 0&3&0\\ 0&0&-2\end{bmatrix}$
$\begin{bmatrix}5&1&1\\ 0&5&1\\ 0&0&5\end{bmatrix}$

i.

If $F=\mathbb{R}$ , which of these matrices are diagonal? Upper triangular? Lower triangular?
ii.

If $F=\mathbb{F}_{5}$ (see Exercise 1.2), how do your answers above change?

[End of Exercise]