Download 16D Multiplicative inverse and solving matrix equations

Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 1 of 10
Chapter 16 Matrices
16D Multiplicative inverse and solving matrix
equations
Recall from arithmetic that any number multiplied by its reciprocal (multiplicative inverse) results in 1. For example,
.
Interactivity
int-0197
Solving matrix equations
Now, consider the matrix multiplication below.
Notice that the answer is the identity matrix (I).
This means that one matrix is the multiplicative inverse of the other. In matrices, we use the symbol A−l to denote the
multiplicative inverse of A. A must be a square matrix.
If AA−1 = A−1A = I, then A−1 is called the multiplicative inverse of A.
Finding the inverse of a square 2 × 2 matrix
In matrices, we are often given a square 2 × 2 matrix and are asked to determine its inverse. The following notes
display how this is achieved.
Let A represent a matrix of the form
.
−
1. Swap the elements on the main diagonal and multiply the elements on the other diagonal by 1. This results in
the matrix
.
2. Multiply this matrix by
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 2 of 10
Note: The value of ad − be is known as the determinant of matrix A. It is commonly written as det A or |A|.
WORKED EXAMPLE 10
Calculate the determinants of the following matrices.
THINK
WRITE
det A = ad − bc
1
For a matrix of the form
, the determinant is given by ad − bc.
2 Calculate the determinant for each of the three matrices.
Singular matrices
Notice that the determinant of matrix C in Worked example 10 was 0. For any matrix that has a determinant of 0, it is
impossible for an inverse to exist. This is because is undefined. A matrix with a determinant equal to 0 is called a
singular matrix.
WORKED EXAMPLE 11
, find its inverse, A−1.
If
Tutorial
int-0513
Worked example 11
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 3 of 10
THINK
WRITE/DISPLAY
Method 1: Using the rule
1 Calculate the determinant of A.
(If the determinant is equal to 0,
the inverse will not exist.)
2 Use the rule to find the inverse.
That is,
.
Swap the elements on the main
diagonal and multiply the
elements on the other diagonal
−
by 1.
Note: Fractional scalars can be left outside the matrix unless they give
whole numbers when multiplied by each element.
Multiply this matrix by
.
Method 2: Using a CAS
calculator
1 Open a Calculator page and
define the matrix A. To do this,
complete the entry line as:
Define
Then press ENTER
.
2 To find the determinant,
complete the entry line as:
det(a)
Then press ENTER
.
3 To find the inverse of matrix A,
complete the entry line as:
a−1
Then press ENTER
.
Note: Some CAS calculators will
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 4 of 10
show the matrix elements as
fractions. Where possible, you
should move fractional scalars
common to each element
outside the matrix (similar to
factorising algebraic
expressions).
4 Write the answer.
For matrices of a higher order than 2 × 2, for example, 3 × 3, 4 × 4, and so on, finding the inverse (and determinant)
is more difficult and the graphics calculator is required. The calculator method involved is identical to that shown in
the worked example above.
Further matrix equations
Recall from algebra that to solve an equation in the form 4x = 9, we need to divide both sides by 4 (or multiply both
sides by ) to obtain the solution
.
A matrix equation of the type AX = B is solved in a similar manner. Both sides of the equation are multiplied by A−1.
Since the order of multiplying matrices is important, we must be careful of the position of the inverse (remember that
the products of AX and XA are different).
Solving for X in the following situations:
1. For AX = B
2. For XA = B
In summary:
1. If AX = B, then X = A−1B
2. If XA = B, then X = B A
−1
WORKED EXAMPLE 12
For the given matrices
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 5 of 10
solve for the unknown matrix X if:
a AX = B
b
.
THINK
WRITE
a 1 We are required to pre-multiply by A−1 to get matrix X
by itself.
a
.
2 Find A−1. First calculate the determinant. Then swap
the elements on the leading diagonal of A and multiply
−
the elements on the other diagonal by 1.
3 Write the equation to be solved and substitute in the
matrices.
4 Calculate the product of A−1 and B. Multiply each
element by the fractional scalar as they will all result in
whole numbers.
b 1 We are required to pre-multiply by A−1 to get matrix X
by itself.
b
2 Use A−1 from part a. Calculate the product of A−1 and
. Multiply each element by the fractional scalar.
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 6 of 10
REMEMBER
1. The inverse of matrix Ais commonly written as A−1.
2. If A A−1 = A−1 A = I, then A−1 is called the multiplicative inverse of A.
3. For a matrix of the form
:
(a)
(b) The value of ad − be is known as the determinant of matrix A. It is commonly written as det A or |A|.
4. A matrix with a determinant equal to 0 is called a singular matrix. For any matrix that has a determinant of
0, it is impossible for an inverse to exist.
5. When solving matrix equations:
(a) If AX = B, then X = A−1 B (Pre-multiply both sides by A−1)
−1
(b) If XA = B, then X = B A
−1
(Post-multiply both sides by A )
EXERCISE 16D Multiplicative inverse and solving
matrix equations
1 WE10
Calculate the determinants of the following matrices.
a
b
c
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 7 of 10
d
2 State which matrices from question 1 will not have an inverse. Give a reason for your answer.
3 WE11 If
, find its inverse, C−1.
EXAM TIP
A common incorrect answer is the original matrix raised to a power of negative one.
4 Calculate the inverse matrix for each matrix (where possible) in question 1.
5 Given the two matrices below, show that A and B are inverses of each other.
6 MC If
, then T−1 is equal to:
A
B
C
D
E
7 MC If
, then P−1 could be:
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 8 of 10
A
B
C
D
E
8 For the matrix
a calculate the determinant
b state the inverse in the form
.
9 Use a CAS calculator to find
i the determinant
ii the inverse matrix
for each of the following matrices.
a
b
c
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and solvi... Page 9 of 10
d
e
f
g
h
i
j
10 WE12 For the given matrices
solve for the unknown matrix X if
a AX = B
b XA = B
c XC = A
d
.
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Maths Quest 12 Further Mathematics 3E TI 2.0 ED - 16 Matrices - 16D Multiplicative inverse and so... Page 10 of 10
11 If
and
, solve the following matrix equations.
a
b
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