4.03a - The language of matrices | OCR A-Level Further Mathematics A

A matrix keeps numbers in positions, so its shape and the order of its entries matter as much as the entries themselves. Learn to read real and complex matrices, recognise the special matrices, compare matrices and take a transpose. This vocabulary will let you describe calculations precisely before you begin them.

Shape and entries

Can two arrays contain exactly the same numbers and still be different mathematical objects? Compare

\mathbf{A}=\begin{pmatrix}2&5&-1\\4&0&3\end{pmatrix},\qquad \mathbf{B}=\begin{pmatrix}2&5\\-1&4\\0&3\end{pmatrix}

Both contain the same six numbers. Yet the first has two rows and the second has three. A matrix records an arrangement, not a bag of numbers. The surprise is that this small distinction eventually decides whether two calculations can even be performed.

Matrix and dimensions

A matrix is a rectangular array of entries arranged in rows and columns. Its dimensions or order are $m$ by $n$ , where $m$ is the number of rows and $n$ the number of columns. The entry $a_{ij}$ is in row $i$ , column $j$ . For example, $\mathbf{A}$ above is $2$ by $3$ , and $a_{23}=3$ .

Read across a row and down a column. A one-row matrix is a row matrix; a one-column matrix is a column matrix. A column vector is therefore also a matrix. The order of $\begin{pmatrix}7&-2&6\end{pmatrix}$ is $1$ by $3$ , while its column version has order $3$ by $1$ . Write matrix entries without commas: $\begin{pmatrix}7&-2\end{pmatrix}$ is a row matrix, whereas $(7,-2)$ normally names a point.

A square matrix has equal numbers of rows and columns. A square matrix of order $3$ is $3$ by $3$ , with nine entries. Matrices whose row and column counts differ are called rectangular matrices. All matrix arrays are rectangular in shape; in the square/rectangular classification, “rectangular” means non-square.

You already know

$i^2=-1$ , and real numbers are complex numbers with imaginary part zero. An entry such as $3-2i$ is one number, so it occupies one position.

A real matrix has only real entries. A matrix with complex entries is a complex matrix; real matrices are special cases. For example,

\mathbf{C}=\begin{pmatrix}1+i&2\\-3i&0\end{pmatrix}

is a complex $2$ by $2$ matrix. The $1+i$ is one entry, not two columns. None of this terminology is supplied in the formula booklet: learn it.

Key idea

Count rows first, then columns; an entry keeps both its value and its position.

This first check separates dimensions from the number of entries.

Equal matrices

Equality compares positions, so matrices are equal only when their shapes match and every corresponding entry matches.

Equal matrices

Two matrices are equal if they have the same dimensions and equal entries in every corresponding position. For example, $\begin{pmatrix}2&3\end{pmatrix}=\begin{pmatrix}2&\sqrt{9}\end{pmatrix}$ , but $\begin{pmatrix}2&3\end{pmatrix}\ne\begin{pmatrix}2\\3\end{pmatrix}$ . This equality rule is not in the formula booklet: learn it.

With unknown entries, a matrix equation is a compact collection of ordinary equations. Use all the positions: one matching entry does not establish equality.

Worked example 1: comparing real entries

Question. Find $x$ and $y$ if $\begin{pmatrix}x+2&5\\-1&2y\end{pmatrix}=\begin{pmatrix}7&5\\-1&-6\end{pmatrix}$ .

Plan. Equate entries in the same positions and solve the resulting equations.

\begin{aligned} x+2&=7 && \text{top-left entries}\\ 2y&=-6 && \text{bottom-right entries}\\ x&=5,\qquad y=-3 && \text{solve} \end{aligned}

Why this step? Matrix equality preserves position; the top-left entry cannot be compared with the bottom-right one.

Answer. $x=5$ and $y=-3$ .

Check. Substitution gives $\begin{pmatrix}7&5\\-1&-6\end{pmatrix}$ on both sides, including the two entries without unknowns.

The second example changes the entry type: an entry now contains complex numbers. Compare positions first, then equate real and imaginary parts inside that position.

Worked example 2: comparing complex entries

Question. The numbers $a$ and $b$ are real. Find them if $\begin{pmatrix}a+bi&2\\-i&4\end{pmatrix}=\begin{pmatrix}3-5i&2\\-i&4\end{pmatrix}$ .

Plan. Equate the top-left entries, then compare their real and imaginary parts.

\begin{aligned} a+bi&=3-5i && \text{same position}\\ a&=3,\qquad b=-5 && \text{equate real and imaginary parts} \end{aligned}

Why this step? The unknowns are real, so equality of the complex entries gives two real conditions.

Answer. $a=3$ and $b=-5$ .

Check. The other three corresponding entries already agree, so the whole matrices are equal.

Every position must agree, even when no unknown appears there. Rearranging entries can destroy equality.

Watch out

Wrong: $\begin{pmatrix}1&2\\3&4\end{pmatrix}=\begin{pmatrix}1&3\\2&4\end{pmatrix}$ because the numbers are the same. The top-right entries are $2$ and $3$ , so the matrices are unequal.

Use exactly the same comparison in your turn.

The transpose

Transposing a matrix exchanges its rows with its columns without changing any entry's value.

Transpose

The transpose of $\mathbf{A}$ , written $\mathbf{A}^{\mathrm T}$ , is formed by making each row of $\mathbf{A}$ the corresponding column of $\mathbf{A}^{\mathrm T}$ . Thus $(\mathbf{A}^{\mathrm T})_{ij}=a_{ji}$ . An $m$ by $n$ matrix has an $n$ by $m$ transpose. These results are not in the formula booklet: learn them.

The first row becomes the first column, the second row becomes the second column, and so on. Transposing twice returns each entry to its starting position, so $(\mathbf{A}^{\mathrm T})^{\mathrm T}=\mathbf{A}$ , also not a supplied result.

Worked example 3: transposing a rectangle

Question. Write down the dimensions and transpose of $\mathbf{A}=\begin{pmatrix}2&-1&4\\0&3&5\end{pmatrix}$ .

Plan. Count the rows and columns, then turn each row into a column.

\begin{aligned} \operatorname{order}(\mathbf{A})&=2\text{ by }3 && \text{two rows, three columns}\\ \mathbf{A}^{\mathrm T}&=\begin{pmatrix}2&0\\-1&3\\4&5\end{pmatrix} && \text{rows become columns} \end{aligned}

Why this step? Keeping the first row together as the first column prevents a reversal of the entry order.

Answer. $\mathbf{A}$ is $2$ by $3$ ; its displayed transpose is $3$ by $2$ .

Check. Transposing the answer gives $\begin{pmatrix}2&-1&4\\0&3&5\end{pmatrix}$ again.

The second example changes only the entry type. A transpose moves complex entries; it does not conjugate them.

Worked example 4: transposing complex entries

Question. Write down $\mathbf{C}^{\mathrm T}$ , where $\mathbf{C}=\begin{pmatrix}1+i&2&-i\\3&4i&5\end{pmatrix}$ .

Plan. Use the same row-to-column rule, leaving every sign unchanged.

\begin{aligned} (1+i,\ 2,\ -i)&\longrightarrow\text{ first column} && \text{move the first row}\\ \mathbf{C}^{\mathrm T}&=\begin{pmatrix}1+i&3\\2&4i\\-i&5\end{pmatrix} && \text{move the second row too} \end{aligned}

Why this step? Transpose refers to positions, so the imaginary parts do not change sign.

Answer. $\mathbf{C}^{\mathrm T}=\begin{pmatrix}1+i&3\\2&4i\\-i&5\end{pmatrix}$ .

Check. The original $c_{23}=5$ becomes entry $(3,2)$ , still equal to $5$ .

The transpose records the same entries in new positions. There is no arithmetic to perform on the entries themselves.

Watch out

Wrong: changing $1+i$ into $1-i$ while taking a transpose. That takes a conjugate as well. Right: move $1+i$ unchanged to its new position.

The next check keeps the shape and varies the entries.

Special matrices and conformability

Some matrix names describe their entries; another describes whether a particular operation is possible.

Zero and identity matrices

A zero matrix, also called a null matrix, has every entry zero, and is written $\mathbf{0}$ , with dimensions specified when needed. An identity matrix $\mathbf{I}$ is square, with $1$ in every main-diagonal position and $0$ elsewhere. For example,

\mathbf{0}_{2\times3}=\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix},\qquad \mathbf{I}_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}

The main diagonal runs from top left to bottom right. A square matrix with zeros away from that diagonal is diagonal; the identity is one example, but $\begin{pmatrix}2&0\\0&3\end{pmatrix}$ is diagonal and is not an identity. The zero matrix need not be square. The names and forms of these matrices are not given in the formula booklet.

A determinant is a single number associated with a square matrix, written $\det\mathbf{A}$ or $\lvert\mathbf{A}\rvert$ . Thus $\begin{vmatrix}a&b\\c&d\end{vmatrix}$ denotes a number, whereas $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ denotes a matrix. A non-square matrix has no determinant. You will learn how to calculate and interpret determinants later; neither the meaning nor the $2$ by $2$ calculation rule is supplied in the formula booklet.

Conformable

Matrices are conformable for an operation when their dimensions allow that operation. For addition or subtraction, the dimensions must match. For the product $\mathbf{AB}$ , the number of columns of $\mathbf{A}$ must equal the number of rows of $\mathbf{B}$ . If $\mathbf{A}$ is $m$ by $n$ and $\mathbf{B}$ is $n$ by $p$ , their product is $m$ by $p$ . These dimension rules are not in the formula booklet: learn them.

For example, a $2$ by $3$ matrix and a $3$ by $4$ matrix cannot be added, but can be multiplied in that order: their product is $2$ by $4$ . Reversing them would require $4=2$ , so that reverse product is undefined. “Conformable” needs the operation and, for multiplication, the order.

Key idea

Before calculating, ask whether the shapes fit the operation. Identical entries do not compensate for different dimensions.

The opening arrays are different: one is $2$ by $3$ , the other $3$ by $2$ . You can now say why, and which operations their shapes permit.

Quick Recap

Dimensions mean rows by columns; a square matrix has the same number of each.
Equal matrices have the same shape and equal corresponding entries, including complex entries.
A transpose exchanges rows and columns without conjugating; transposing twice returns the starting matrix.
A zero matrix has all entries zero, an identity has diagonal ones and other entries zero, and a determinant is a number attached to a square matrix.
Conformability depends on the operation: identical shapes for addition; matching inner dimensions for multiplication.

The opening arrays contain the same numbers but have different shapes, so they are different matrices.

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4.03a - The language of matrices

Shape and entries

Equal matrices

The transpose

Special matrices and conformability

Quick Recap

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