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Determinant and Inverse of 3 by 3 Matrices!
Starter
1. Find the matrix such that .
2. If is the matrix , show that . Hence find .
Hint: Remember
Notes
Determinant of a 3 by 3 matrix
The determinant of a matrix uses the determinant of a matrix three times.
Let
Here is the formula to find the determinant of a matrix:
To understand it, we consider each part separately:
This matrix shows the signs that go with each determinant:
Alternative formula:
X
(
1 3
2 4
)
X
−1
=
(
−2 9
−2 14
)
A
(
7 4
6 3
)
A
2
− 10A − 3I = 0
A
−1
3!by!3
2!by!2
M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
3!by!3
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
− b
1
a
2
c
2
a
3
c
3
+ c
1
a
2
b
2
a
3
b
3
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
−b
1
a
2
c
2
a
3
c
3
+ c
1
a
2
b
2
a
3
b
3
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
−b
1
a
2
c
2
a
3
c
3
+c
1
a
2
b
2
a
3
b
3
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
− b
1
a
2
c
2
a
3
c
3
+c
1
a
2
b
2
a
3
b
3
2!by!2
(
+ − +
− + −
+ − +
)
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
− a
2
b
1
c
1
b
3
c
3
+ a
3
b
1
c
1
b
2
c
2
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E.g. 1 Find the determinant of the matrix .
Key facts to help you calculate the determinant of a matrix
Here are a few shortcuts to help speed up the calculation of the determinant. The aim is to simplify
the calculation, usually by getting as many zeros in the top row as possible.
When two rows or columns are swapped over, the determinant changes sign.
E.g.
If a row or column has a common factor, it is a factor of the determinant.
E.g.
To any row (or column) can be added multiples of any other row (or column) without
altering the value of the determinant
E.g. Consider .
gives:
If a matrix has two identical rows or columns then the determinant is zero.
E.g.
The determinant of a matrix is equal to the determinant of the transposed matrix i.e.
. Therefore, if there is a zero in the first column, transpose and
calculate the determinant or use the alternative formula.
E.g.
(
1 −2 3
1 −1 2
−2 4 −5
)
3!by!3
1 6 7
3 0 0
−8 2 3
= −
3 0 0
1 6 7
−8 2 3
= − 3
6 7
2 3
= . . .
7 2 −5
4 12 6
1 −8 4
= 2
7 2 −5
2 6 3
1 −8 4
5 2 2
−2 1 1
8 −6 3
R
1
− 2R
2
5 2 2
−2 1 1
8 −6 3
=
9 0 0
−2 1 1
8 −6 3
= 9
1 1
−6 3
= . . .
1 1 1
1 1 1
4 −9 7
= 0
det!M = det!M
T
2 4 1
0 −5 3
0 7 6
=
2 0 0
4 −5 7
1 3 6
= 2
−5 7
3 6
= . . .
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E.g. 2 Using the shortcuts above, find the determinants of these matrices:
(a) (b) (c)
Working: (a) Transpose:
:
Inverse of a 3 by 3 process
Non-calculator
Matrix of cofactors
The first step to find the inverse of a matrix by hand is to calculate the matrix of cofactors.
The cofactor, , of is the determinant left after the the row and column in which lies
have been removed.
N.B. Remember to use the matrix of signs:
Here are the complete cofactors:
Success Criteria — finding the inverse of a 3 by 3 matrix (non-calculator)
Let
1. Find the determinant of the matrix, .
2. Find the matrix of cofactors
3. Transpose the matrix of cofactors to get
(
1 −2 3
1 −1 2
−2 4 −5
)
(
1 1 1
−1 0 1
−2 −2 0
)
(
2 −1 3
1 2 1
3 −4 5
)
1 −2 3
1 −1 2
−2 4 −5
=
1 1 −2
−2 −1 4
3 2 −5
R
1
+ R
2
−1 0 2
−2 −1 4
3 2 −5
−1 0 2
−2 −1 4
3 2 −5
= − 1
−1 4
2 −5
+ 2
−2 −1
3 2
= 3 + 2 × (−1) = 1
3!by!3
A
1
a
1
2!by!2
a
1
(
+ − +
− + −
+ − +
)
A
1
=
b
2
c
2
b
3
c
3
B
1
= −
a
2
c
2
a
3
c
3
C
1
=
a
2
b
2
a
3
b
3
A
2
= −
b
1
c
1
b
3
c
3
B
2
=
a
1
c
1
a
3
c
3
C
2
= −
a
1
b
1
a
3
b
3
A
3
=
b
1
c
1
b
2
c
2
B
3
= −
a
1
c
1
a
2
c
2
C
3
=
a
1
b
1
a
2
b
2
M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
det!M
C =
A
1
B
1
C
1
A
2
B
2
C
2
A
3
B
3
C
3
C
T
=
A
1
A
2
A
3
B
1
B
2
B
3
C
1
C
2
C
3
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4. Divide the matrix of cofactors by the determinant:
From this we can see that
N.B. If the matrix is singular i.e. , then the inverse of the matrix does not exist.
E.g. 3 Find the inverse of the matrix
Working:
Matrix of cofactors is so
E.g. 4 Find the inverse of the matrix .
Hint: You have already found its determinant in E.g. 2
Using a calculator to find the inverse of a matrix
Menu >> 4 : Matrix >> Press 1–4 to select Matrix A—D >> Choose the number of rows >> Choose
the number of columns >> Enter the elements of the matrix >> OPTN >> 3: Matrix Calc >> OPTN
Finding the determinant >> (Down arrow) >> 2 : Determinant >> OPTN >> (Choose the matrix)
>> Press >> Press
Finding the inverse >> Press 3–6 to select Mat A—D >> Press
Video: Using a calculator to find the determinant and/or the inverse of matrices
E.g. 5 Use your calculator to find the inverse of these matrices:
(a) (b)
Working: (a)
M
−1
=
1
det!M
C
T
=
1
det!M
A
1
A
2
A
3
B
1
B
2
B
3
C
1
C
2
C
3
M × C
T
=
(
det!M 0 0
0 det!M 0
0 0 det!M
)
det!M = 0
M =
(
4 1 −1
3 −1 2
4 0 −3
)
det!M = 25
C =
(
3 17 4
3 −8 4
1 −11 −7
)
C
T
=
(
3 3 1
17 −8 −11
4 4 −7
)
M
−1
=
1
25
(
3 3 1
17 −8 −11
4 4 −7
)
(
1 −2 3
1 −1 2
−2 4 −5
)
3!by!3
)
=
x
−1
(
1 −2 −3
2 −1 −4
3 −3 −5
)
(
1 −2 −1
2 1 5
3 −2 3
)
−
7
6
−
1
6
5
6
−
1
3
2
3
−
1
3
−
1
2
−
1
2
1
2
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E.g. 6 Using your calculator, solve for the equation .
Video: Determinant of 3 by 3 matrices
Video A: Inverse of 3 by 3 matrices
Video B: Inverse of 3 by 3 matrices
Video: Using a calculator to find the determinant and/or the inverse of matrices
Solutions to Starter and E.g.s
Exercise
p27 1E Qu 1i, 2i, 3i, 4i, 5-9
Summary
A singular matrix is when the determinant equals zero i.e.
When two rows or columns are swapped over, the determinant changes sign.
If a row or column has a common factor, it is a factor of the determinant.
To any row (or column) can be added multiples of any other row (or column) without
altering the value of the determinant
If a matrix has two identical rows or columns then the determinant is zero.
The determinant of a matrix is equal to the determinant of the transposed matrix i.e.
. Therefore, if there is a zero in the first column, transpose and
calculate the determinant or use the alternative formula.
Cofactors:
X
(
1 3 2
4 0 −1
2 3 −3
)
X =
0 −6 3
21 6 −9
−9 5 −4
det!M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
= a
1
b
2
c
2
b
3
c
3
− b
1
a
2
c
2
a
3
c
3
+ c
1
a
2
b
2
a
3
b
3
det!M = 0
det!M = det!M
T
A
1
=
b
2
c
2
b
3
c
3
B
1
= −
a
2
c
2
a
3
c
3
C
1
=
a
2
b
2
a
3
b
3
A
2
= −
b
1
c
1
b
3
c
3
B
2
=
a
1
c
1
a
3
c
3
C
2
= −
a
1
b
1
a
3
b
3
A
3
=
b
1
c
1
b
2
c
2
B
3
= −
a
1
c
1
a
2
c
2
C
3
=
a
1
b
1
a
2
b
2
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Success Criteria — finding the inverse of a 3 by 3 matrix (non-calculator):
Let
1. Find the determinant of the matrix, .
2. Find the matrix of cofactors
3. Transpose the matrix of cofactors to get
4. Divide the matrix of cofactors by the determinant:
Using a calculator to find the inverse of a matrix:
Menu >> 4 : Matrix >> Press 1–4 to select Matrix A—D >> Choose the number of rows >> Choose
the number of columns >> Enter the elements of the matrix >> OPTN >> 3: Matrix Calc >> OPTN
Finding the determinant >> (Down arrow) >> 2 : Determinant >> OPTN >> (Choose the matrix)
>> Press >> Press
Finding the inverse >> Press 3–6 to select Mat A—D >> Press
M =
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
det!M
C =
A
1
B
1
C
1
A
2
B
2
C
2
A
3
B
3
C
3
C
T
=
A
1
A
2
A
3
B
1
B
2
B
3
C
1
C
2
C
3
M
−1
=
1
det!M
C
T
=
1
det!M
A
1
A
2
A
3
B
1
B
2
B
3
C
1
C
2
C
3
3!by!3
)
=
x
−1
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