Matrices and Matrix Operations

Num­bers arranged in a table#

A matrix is just num­bers arranged neatly in rows and columns. You have seen such tables many times: the sales of three prod­ucts in two shops, or the coef­fi­cients of a sys­tem of equa­tions. We will name the dif­fer­ent types of matri­ces and learn to add them, mul­ti­ply them by num­bers, mul­ti­ply them together and trans­pose them. Keep one warn­ing in mind through­out: with matri­ces, the order of mul­ti­pli­ca­tion mat­ters.

Matri­ces#

A matrix of order $m \times n$ has $m$ rows and $n$ columns, and $a_{ij}$ is the entry sit­ting in row $i$, col­umn $j$. We write $A = [a_{ij}]$.

The com­mon types. Col­umn ($n = 1$), row ($m = 1$), square ($m = n$), diag­o­nal (square, zero off the diag­o­nal), scalar (diag­o­nal with equal diag­o­nal entries), iden­tity $I$ (scalar with $1$s), zero $O$.

Oper­a­tions#

• Sum (same order): add cor­re­spond­ing entries. This is com­mu­ta­tive and asso­cia­tive, and $A + O = A$.
• Scalar mul­ti­ple: $kA = [k a_{ij}]$; $k(A + B) = kA + kB$.
• Prod­uct: for $A$ of order $m \times n$ and $B$ of order $n \times p$, $AB$ is $m \times p$ with $\displaystyle (AB)_{ik} = \sum_j a_{ij} b_{jk}$. In words: run along a row of $A$ and down a col­umn of $B$, mul­ti­ply pairs and add.

What still works, and what does not. $(AB)C = A(BC)$; $A(B + C) = AB + AC$; $AI = IA = A$. But here is a sur­prise: $AB = O$ can hap­pen even when $A, B \ne O$. For exam­ple, $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = O$.

Trans­pose#

The trans­pose $A^T$ turns rows into columns: $(A^T)_{ij} = a_{ji}$. Its rules are easy to remem­ber: $(A^T)^T = A$, $(A + B)^T = A^T + B^T$, $(kA)^T = kA^T$, and, with the order reversed, $(AB)^T = B^T A^T$.

Try these your­self#

1. Con­struct the $3 \times 2$ matrix with $a_{ij} = \lvert 2i - j \rvert$.
2. Find $x, y, z$: $\begin{bmatrix} x + y & 2 \\ 5 & xz \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ 5 & 12 \end{bmatrix}$ with $x = 4$.
3. $A = \begin{bmatrix} 1 & 2 & -3 \\ 0 & 4 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 5 \end{bmatrix}$. Find $2A - 3B$.
4. Find $X$ if $3X + \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 7 & -4 \\ 9 & 5 \end{bmatrix}$.
5. Com­pute $\begin{bmatrix} 1 & 3 \\ -2 & 0 \end{bmatrix}\begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix}$ and $\begin{bmatrix} 2 & 1 & 0 \end{bmatrix}\begin{bmatrix} 3 \\ -1 \\ 7 \end{bmatrix}$.
6. For $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$, show $A^2 - 4A + I = O$.
7. Ver­ify $(AB)^T = B^TA^T$ for $A = \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 1 \end{bmatrix}$.
8. If $A$ is $3 \times 4$ and $B$ is $4 \times 2$, what is the order of $AB$? Is $BA$ defined?
9. A fac­tory makes items X and Y using $2$ and $3$ hours of labour and $4$ and $1$ kg of mate­r­ial. Labour costs ₹$150$ per hour and mate­r­ial ₹$60$ per kg. Use matri­ces to find the cost of each item.
10. Find two non-zero $2 \times 2$ matri­ces whose prod­uct is $O$.

Answers to check against#