Solving matrix equations and systems | Year 11

01 / LEARN

The idea, step by step

Build the coefficient matrix

Write the equations with variables in the same order. The coefficients form A, the unknowns form a column X, and the constants form a column b. Then $A X = b$ represents both equations at once.

A missing variable has coefficient zero. Keeping that zero in the matrix preserves the equation and the positions of the variables.

Undo on the left

For invertible A, multiply both sides on the left by $A^{- 1}$ to obtain $X = A^{- 1} b$ . Matrix order matters: $b A^{- 1}$ is usually not even defined.

Check the calculated pair in both original equations. For a matrix unknown in $A X = B$ , the same left multiplication gives $X = A^{- 1} B$ . If the equation is $X A = B$ , undo on the right instead.

A common trap

Writing $X = b A^{- 1}$ for $A X = b$ .

Instead: Multiply on the side adjacent to the coefficient matrix so that the inverse and matrix meet.

02 / WORKED EXAMPLES

See the thinking

Example 1

Solve a pair

Solve $2 x + y = 7$ and $3 x + 2 y = 12$ .

$A = (\begin{array}{cccccccccc} 2 & 1 \\ 3 & 2 \end{array})$ , $b = (\begin{array}{cccccccccc} 7 \\ 12 \end{array})$ .
$A^{- 1} = (\begin{array}{cccccccccc} 2 & - 1 \\ - 3 & 2 \end{array})$ .
$X = A^{- 1} b = (\begin{array}{cccccccccc} 2 \\ 3 \end{array})$ ; substitution gives 7 and 12.

x = 2

y = 3

Example 2

Solve for a whole matrix

Solve $(\begin{array}{cccccccccc} 1 & 1 \\ 0 & 1 \end{array}) X = (\begin{array}{cccccccccc} 3 & 5 \\ 2 & 4 \end{array})$ .

The inverse on the left is $(\begin{array}{cccccccccc} 1 & - 1 \\ 0 & 1 \end{array})$ .
Multiply that inverse by the right-hand matrix.

X = (\begin{array}{cccccccccc} 1 & 1 \\ 2 & 4 \end{array})

Check your understanding

If $X A = B$ and A is invertible, what is X?

Show answer

$X = B A^{- 1}$ .

Undo A by multiplying both sides on the right.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write $3 x - y = 4$ , $2 x + 5 y = 1$ as one matrix equation.
2 marks
Solve $x + y = 5$ , $x - y = 1$ .
2 marks
Solve $2 x + 3 y = 1$ , $x + 2 y = 2$ .
2 marks
Solve $(\begin{array}{cccccccccc} 2 & 0 \\ 0 & 3 \end{array}) X = (\begin{array}{cccccccccc} 8 \\ 9 \end{array})$ .
2 marks
Solve $X (\begin{array}{cccccccccc} 1 & 2 \\ 0 & 1 \end{array}) = (\begin{array}{cccccccccc} 3 & 7 \\ 1 & 4 \end{array})$ .
2 marks
A ticket model gives $2 a + b = 17$ and $a + 2 b = 19$ . Find a and b and check both equations.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $3 x - y = 4$ , $2 x + 5 y = 1$ as one matrix equation.
2 marks
Keep x first and y second in both rows.
$(\begin{array}{cccccccccc} 3 & - 1 \\ 2 & 5 \end{array}) (\begin{array}{cccccccccc} x \\ y \end{array}) = (\begin{array}{cccccccccc} 4 \\ 1 \end{array})$ .
Solve $x + y = 5$ , $x - y = 1$ .
2 marks
Add the equations to get $2 x = 6$ .
Then $y = 5 - 3 = 2$ ; the matrix method gives the same result.
$x = 3$ , $y = 2$ .
Solve $2 x + 3 y = 1$ , $x + 2 y = 2$ .
2 marks
The determinant is 1.
The inverse $(\begin{array}{cccccccccc} 2 & - 3 \\ - 1 & 2 \end{array})$ applied to ${(1, 2)}^{T}$ gives ${(- 4, 3)}^{T}$ .
$x = - 4$ , $y = 3$ .
Solve $(\begin{array}{cccccccccc} 2 & 0 \\ 0 & 3 \end{array}) X = (\begin{array}{cccccccccc} 8 \\ 9 \end{array})$ .
2 marks
Undo the two diagonal scales.
$X = (\begin{array}{cccccccccc} 4 \\ 3 \end{array})$ .
Solve $X (\begin{array}{cccccccccc} 1 & 2 \\ 0 & 1 \end{array}) = (\begin{array}{cccccccccc} 3 & 7 \\ 1 & 4 \end{array})$ .
2 marks
Multiply the right-hand matrix on the right by $(\begin{array}{cccccccccc} 1 & - 2 \\ 0 & 1 \end{array})$ .
$X = (\begin{array}{cccccccccc} 3 & 1 \\ 1 & 2 \end{array})$ .
A ticket model gives $2 a + b = 17$ and $a + 2 b = 19$ . Find a and b and check both equations.
2 marks
Eliminate to obtain $3 a = 15$ .
Then $b = 17 - 10 = 7$ ; the second equation gives $5 + 14 = 19$ .
$a = 5$ , $b = 7$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

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