Year 11 General Matrices And Matrix Arithmetic

Adding And Subtracting Matrices

5 practice questions 1 video lesson Theory + worked examples

Theory

Two matrices can be added or subtracted only if they have the same order. The operation is done element by element: $(A + B)_{i j} = A_{i j} + B_{i j}$ . Matrix addition is commutative and associative; subtraction is not. Solve matrix equations like $A + X = C$ by rearranging just as with numbers.

Two matrices can be added or subtracted only if they have the same order — the same number of rows AND the same number of columns. The result is a matrix of that same order, with each entry computed by adding (or subtracting) the corresponding entries:

$(A + B)_{i j} = A_{i j} + B_{i j} (A - B)_{i j} = A_{i j} - B_{i j}$

The zero matrix $0$ of a given order has every entry equal to 0. It is the identity for matrix addition: $A + 0 = A$ for any $A$ of the same order.

Matrix addition is commutative ( $A + B = B + A$ ) and associative ( $(A + B) + C = A + (B + C)$ ). Matrix subtraction is neither.

The first diagram shows how addition works entry-by-entry. The second shows that the operation is only defined when the matrices have the same order.

Add the entries in matching positions to build the result, entry by entry.

Two

2 \times 3

matrices can be added. A

2 \times 3

plus a

3 \times 2

is undefined.

The rules are simple element-wise definitions.

Addition

$(A + B)_{i j} = A_{i j} + B_{i j}$

(A + B)_{i j} = A_{i j} + B_{i j}

Subtraction

$(A - B)_{i j} = A_{i j} - B_{i j}$

(A - B)_{i j} = A_{i j} - B_{i j}

Key properties

Property	Statement
Commutative (addition)	$A + B = B + A$
Associative (addition)	$(A + B) + C = A + (B + C)$
Zero matrix is identity	$A + 0 = A$
Subtraction is not commutative	$A - B \neq B - A$ in general

Solving matrix equations. If A+X=C, subtract A from both sides to get X=C−A. The balance rules from ordinary algebra carry over directly.

How to add or subtract two matrices

Check the orders match. Both matrices must have the same number of rows AND the same number of columns. If they don't, the operation is undefined.
Set up a result matrix of the same order.
Compute each entry by adding (or subtracting) the corresponding entries from the two originals.
Watch the signs, especially when subtracting negative entries: $- 3 - (- 7) = - 3 + 7 = 4$ .

Solving $A + X = C$ for an unknown matrix

Rearrange to isolate $X$ : $X = C - A$ .
Subtract matrix $A$ from matrix $C$ , entry by entry.
State the resulting matrix $X$ .

Solving for a missing entry

Use the addition rule on each entry. Most entries will simplify with no unknowns and just confirm the working.
For the entry containing the unknown, write the equation it gives and solve.

EXAMPLE 1 — STANDARD ADDITION

Find

A + B

where

A = [\begin{matrix} 5 & - 2 \\ - 3 & 8 \end{matrix}]

and

B = [\begin{matrix} - 1 & 7 \\ 6 & - 4 \end{matrix}]

SOLUTION

Both matrices are $2 \times 2$ , so addition is defined. Add entry-by-entry.

Row 1, col 1	$=$	$5 + (- 1) = 4$
Row 1, col 2	$=$	$- 2 + 7 = 5$
Row 2, col 1	$=$	$- 3 + 6 = 3$
Row 2, col 2	$=$	$8 + (- 4) = 4$

$A + B = [\begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix}]$

A + B = [\begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix}]

EXAMPLE 2 — SUBTRACTION WITH NEGATIVES

Find

A - B

where

A = [\begin{matrix} 6 & - 3 \\ 2 & 4 \end{matrix}]

and

B = [\begin{matrix} 2 & - 7 \\ 5 & - 1 \end{matrix}]

SOLUTION

Subtract entry-by-entry. Watch the signs — subtracting a negative becomes adding a positive.

Row 1, col 1	$=$	$6 - 2 = 4$
Row 1, col 2	$=$	$- 3 - (- 7) = - 3 + 7 = 4$
Row 2, col 1	$=$	$2 - 5 = - 3$
Row 2, col 2	$=$	$4 - (- 1) = 4 + 1 = 5$

$A - B = [\begin{matrix} 4 & 4 \\ - 3 & 5 \end{matrix}]$

A - B = [\begin{matrix} 4 & 4 \\ - 3 & 5 \end{matrix}]

EXAMPLE 3 — FIND AN UNKNOWN MATRIX

Find

X

such that

A + X = C

, where

A = [\begin{matrix} 2 & 3 \\ 5 & 1 \end{matrix}]

and

C = [\begin{matrix} 7 & 9 \\ 4 & 8 \end{matrix}]

SOLUTION

Rearrange the equation: $X = C - A$ . Then subtract entry-by-entry.

Row 1, col 1	$=$	$7 - 2 = 5$
Row 1, col 2	$=$	$9 - 3 = 6$
Row 2, col 1	$=$	$4 - 5 = - 1$
Row 2, col 2	$=$	$8 - 1 = 7$

$X = [\begin{matrix} 5 & 6 \\ - 1 & 7 \end{matrix}]$

X = [\begin{matrix} 5 & 6 \\ - 1 & 7 \end{matrix}]

EXAMPLE 4 — FIND A MISSING ENTRY

Find

x

given

[\begin{matrix} 3 & x \\ 5 & 2 \end{matrix}] + [\begin{matrix} 1 & 4 \\ 6 & 6 \end{matrix}] = [\begin{matrix} 4 & 7 \\ 11 & 8 \end{matrix}]

SOLUTION

Match corresponding entries. The only unknown is in row 1, column 2.

$x + 4$	$=$	$7$
$x$	$=$	$3$

Answer: $x = 3$ .

x = 3

Common pitfalls

Orders must match exactly. A

2 \times 3

matrix cannot be added to a

3 \times 2

matrix, even though both have six entries. Same rows AND same columns.

Watch double negatives.

- 3 - (- 7) = - 3 + 7 = 4

, not

- 10

. Subtracting a negative becomes adding a positive.

Subtraction is not commutative.

A - B

and

B - A

are different matrices — every entry has the opposite sign. Always subtract in the order asked.

Match entries one-to-one for equality. For a matrix equation like

[\begin{matrix} 3 & x \\ 5 & 2 \end{matrix}] + [\begin{matrix} 1 & 4 \\ 6 & 6 \end{matrix}] = [\begin{matrix} 4 & 7 \\ 11 & 8 \end{matrix}]

, set up one small equation per entry position. Most just confirm; the unknown gives you the one you need.

Don't add the orders. A

2 \times 3

plus a

2 \times 3

is still a

2 \times 3

— not a

4 \times 6

. The order of the result equals the order of the inputs.

Frequently asked questions

When can two matrices be added or subtracted?

Only when they have exactly the same order — the same number of rows AND the same number of columns. A 2 by 3 matrix cannot be added to a 3 by 2 matrix, even though both contain 6 entries.

How do I add two matrices?

Add element by element. The entry in row i, column j of the result is the sum of the entries in row i, column j of the two original matrices. The result has the same order as the originals.

How does subtraction differ from addition?

Subtraction works the same way, element-by-element, but you subtract the corresponding entries instead of adding them. Watch the signs carefully — subtracting a negative entry gives a positive contribution.

Is matrix addition commutative?

Yes — A plus B equals B plus A for any two matrices of the same order. Matrix addition is also associative, meaning the way you group three or more matrices when adding doesn't change the result.

Is matrix subtraction commutative?

No — A minus B is generally not equal to B minus A. The two results differ by a sign in every entry. Always do the subtraction in the order asked.

How do I solve a matrix equation like A plus X equals C?

Treat X as the unknown and rearrange the equation as you would with ordinary numbers. From A plus X equals C, subtract A from both sides to get X equals C minus A. Then compute the right-hand side element by element.

Video Lesson

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Adding And Subtracting Matrices

📖 Theory

Addition

Subtraction

Key properties

How to add or subtract two matrices

Solving A+X=C for an unknown matrix

Solving for a missing entry

Common pitfalls

Frequently asked questions

🎬 Video Lesson

✏️ Practice Questions

Theory

Solving $A + X = C$ for an unknown matrix

Video Lesson

Practice Questions