SME
Grade 12 Math · Problem type
Solving a system with A⁻¹
Use this for “Solve 3x + 2y = 7 and 5x + 4y = 13 with the inverse matrix.”
Diagram
A is 2 × 2 and B is 2 × 1, so AB is 2 × 1.
Row 1 of A times column 1 of B gives entry (1, 1): c₁₁ = 2 × 7 + (−1) × 13 = 1.
Tap an entry of AB to see its row and column.
This is an example. Type your own values to start a new problem.
a
x
+
b
y
=
p
c
x
+
d
y
=
q
x
=
x
,
y
=
y
The picture writes the system as X = A⁻¹B: the inverse of the coefficients times the right sides gives x and y.
D
Determinant ad − bc
worked out (not typed)
Formulas
D
=
a
×
d
−
b
×
c
2 = 3 × 4 − 2 × 5
x
=
d
×
p
−
b
×
q
D
1
=
4 × 7 − 2 × 13
2
y
=
a
×
q
−
c
×
p
D
2
=
3 × 13 − 5 × 7
2
Assumptions
•
The system is AX = B: A holds the coefficients, X = [[x], [y]] and B the right sides.
•
Multiply both sides by A⁻¹ on the left: X = A⁻¹B.
•
A⁻¹ = (1/D)[[d, −b], [−c, a]] with D = ad − bc, so D = 0 means no single solution.
Step-by-step
WE KNOW
a = 3, b = 2, p = 7, c = 5, d = 4, q = 13
FIND
determinant ad − bc (D), solution x (x), solution y (y)
Step 1 · Find determinant ad − bc (D)
Use D = a × d − b × c
A
-1
needs
the
determinant:
down
the
main
diagonal,
take
away
the
other.
D
=
3
×
4
−
2
×
5
D
=
12
−
10
D
=
2
→
A
-1
=
(
1
2
)[[4,
−2],
[−5,
3]]
Step 2 · Find solution x (x)
Use x = (d × p − b × q) ÷ D
Row
1
of
A
-1
times
B:
d
times
p
,
take
away
b
times
q
,
all
over
D
.
x
=
4 × 7 − 2 × 13
2
x
=
28 − 26
2
x
=
2
2
x
=
1
Step 3 · Find solution y (y)
Use y = (a × q − c × p) ÷ D
Row
2
of
A
-1
times
B:
a
times
q
,
take
away
c
times
p
,
all
over
D
.
y
=
3 × 13 − 5 × 7
2
y
=
39 − 35
2
y
=
4
2
y
=
2
Check
Put the numbers back in. Both sides should match.
2 = 3 × 4 − 2 × 5 ✓
1
=
4 × 7 − 2 × 13
2
✓
2
=
3 × 13 − 5 × 7
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Related lessons
Systems in three variables, matrices and determinants
Main lesson
A matrix times a vector
Use this for “Multiply [[2, 1], [3, 4]] by the column (5, −1).”
Determinant of a 3 × 3 matrix
Use this for “Find the determinant of [[2, 0, 1], [1, 3, 2], [1, 1, 4]].”
Inverse of a 2 × 2 matrix
Use this for “Find the inverse of [[4, 7], [2, 6]].”
Cramer’s rule for two equations
Use this for “Solve 2x + 3y = 13 and x − y = −1 by Cramer’s rule.”