LA3: Matrix Algebra

Chapter 17 LA3: Matrix Algebra

We introduced matrices when discussing systems of linear equations. Recall that an

m \times n

matrix is a rectangular array of numbers, written in the form

(\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}),

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which is sometimes written in the abbreviated form

A = {(a_{i j})}_{m \times n} .

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Note that we usually use a capital letter to denote a matrix.

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In the context of systems of linear equations the main concept was row reducing a matrix to produce equivalent matrices. In different contexts, however, other operations on matrices have proved useful and we shall discuss these operations below. Before doing this, though, we will introduce some new terminology.

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Definition 17.0.1.

Two matrices are said to be equal if they are of the same size and all corresponding entries are equal.

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Example 17.0.2.

Let $A = (\begin{matrix} 5 & 0 & 1 \\ - 3 & \sqrt{2} & \frac{1}{2} \end{matrix}), B = (\begin{matrix} 5 & - 3 \\ 0 & \sqrt{2} \\ 1 & \frac{1}{2} \end{matrix}), C = (\begin{matrix} 5 & 0 & 1 \\ - 3 & a & b \end{matrix}) .$

For what values of $a$ and $b$ does $A = C ?$
For what values of $a$ and $b$ does $B = C ?$

Answer.

$a = \sqrt{2}$ and $b = \frac{1}{2}$
None.

Solution.

Since $A$ and $C$ are of the same size, the matrices will be equal when $a = \sqrt{2}$ and $b = \frac{1}{2} .$
Since $B$ and $C$ are not the same size they can never be equal.