Worked examples
The question
Question: solve 2x + y = 5 and x + y = 3 using the inverse matrix.
2x + y = 5
We want the x and y that make both true.
x + y = 3
1 Write it as AX = B
The numbers in front of x and y make the matrix A.
The letters make X.
The right sides make B.
Row times column gives back the equations.
[2, 1; 1, 1] × [x; y] = [5; 3]
To get X alone, we multiply by the inverse of A: X = A⁻¹B.
AX = B → X = A−1B
Careful!
The inverse goes in front of B.
With matrices, the order matters.
Careful: X = A−1B, not BA−1
2 Find the inverse
First D, the determinant of A.
D = |2, 1; 1, 1|
So D = 2 × 1 − 1 × 1.
D = 2 × 1 − 1 × 1
2 times 1 is 2.
1 times 1 is 1.
D = 2 − 1
2 minus 1 is 1.
So D = 1.
D = 1
Swap the top left and bottom right.
Change the signs of the other two.
Then times 1 over D.
A−1 = 11 × [1, −1; −1, 2]
So A⁻¹ = rows 1, −1 and −1, 2.
A−1 = [1, −1; −1, 2]
3 Multiply
X is A⁻¹ times B.
[x; y] = [1, −1; −1, 2] × [5; 3]
= [1 × 5 + (−1) × 3; −1 × 5 + 2 × 3]
Work them out: x = 2 and y = 1.
[x; y] = [2; 1]
So the answer is (2, 1).
(x, y) = (2, 1)
4 Check
Let's check in every equation.
We put x = 2, y = 1 into the first one.
Equation 1 2(2) + 1 = 4 + 1 = 5
We get 5, the same as the right side.
Now equation 2.
Equation 2 2 + 1 = 3
We get 3, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: write it as AX = B.
Then: find the inverse.
Then: multiply.
Last: check.
▶ Watch it solved
The question
Question: solve x − y = 1 and 2x − y = 4 using the inverse matrix.
x − y = 1
We want the x and y that make both true.
2x − y = 4
1 Write it as AX = B
The numbers in front of x and y make the matrix A.
The letters make X.
The right sides make B.
Row times column gives back the equations.
[1, −1; 2, −1] × [x; y] = [1; 4]
To get X alone, we multiply by the inverse of A: X = A⁻¹B.
AX = B → X = A−1B
Careful!
The inverse goes in front of B.
With matrices, the order matters.
Careful: X = A−1B, not BA−1
2 Find the inverse
First D, the determinant of A.
D = |1, −1; 2, −1|
So D = 1 × (−1) − 2 × (−1).
D = 1 × (−1) − 2 × (−1)
1 times (−1) is −1.
2 times (−1) is −2.
D = −1 − (−2)
Taking away a negative is adding: −1 + 2.
−1 − (−2) = −1 + 2
−1 minus (−2) is 1.
So D = 1.
D = 1
Swap the top left and bottom right.
Change the signs of the other two.
Then times 1 over D.
A−1 = 11 × [−1, 1; −2, 1]
So A⁻¹ = rows −1, 1 and −2, 1.
A−1 = [−1, 1; −2, 1]
3 Multiply
X is A⁻¹ times B.
[x; y] = [−1, 1; −2, 1] × [1; 4]
= [−1 × 1 + 1 × 4; −2 × 1 + 1 × 4]
Work them out: x = 3 and y = 2.
[x; y] = [3; 2]
So the answer is (3, 2).
(x, y) = (3, 2)
4 Check
Let's check in every equation.
We put x = 3, y = 2 into the first one.
Equation 1 3 − 2 = 1
We get 1, the same as the right side.
Now equation 2.
Equation 2 2(3) − 2 = 6 − 2 = 4
We get 4, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: write it as AX = B.
Then: find the inverse.
Then: multiply.
Last: check.
▶ Watch it solved
The question
Question: solve 3x + y = 5 and x + y = 3 using the inverse matrix.
3x + y = 5
We want the x and y that make both true.
x + y = 3
1 Write it as AX = B
The numbers in front of x and y make the matrix A.
The letters make X.
The right sides make B.
Row times column gives back the equations.
[3, 1; 1, 1] × [x; y] = [5; 3]
To get X alone, we multiply by the inverse of A: X = A⁻¹B.
AX = B → X = A−1B
Careful!
The inverse goes in front of B.
With matrices, the order matters.
Careful: X = A−1B, not BA−1
2 Find the inverse
First D, the determinant of A.
D = |3, 1; 1, 1|
So D = 3 × 1 − 1 × 1.
D = 3 × 1 − 1 × 1
3 times 1 is 3.
1 times 1 is 1.
D = 3 − 1
3 minus 1 is 2.
So D = 2.
D = 2
Swap the top left and bottom right.
Change the signs of the other two.
Then times 1 over D.
A−1 = 12 × [1, −1; −1, 3]
So A⁻¹ = rows 1/2, −1/2 and −1/2, 3/2.
A−1 = [1 over 2, −1 over 2; −1 over 2, 3 over 2]
3 Multiply
X is A⁻¹ times B.
[x; y] = [1 over 2, −1 over 2; −1 over 2, 3 over 2] × [5; 3]
= [1 over 2 × 5 + (−1 over 2) × 3; −1 over 2 × 5 + 3 over 2 × 3]
Work them out: x = 1 and y = 2.
[x; y] = [1; 2]
So the answer is (1, 2).
(x, y) = (1, 2)
4 Check
Let's check in every equation.
We put x = 1, y = 2 into the first one.
Equation 1 3(1) + 2 = 3 + 2 = 5
We get 5, the same as the right side.
Now equation 2.
Equation 2 1 + 2 = 3
We get 3, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: write it as AX = B.
Then: find the inverse.
Then: multiply.
Last: check.
▶ Watch it solved
