Inverse of Matrix

MATRICES IT LESSON 1 - INVERSE OF A MATRIX
(SEC 3 ADDITIONAL MATHS)

Software: LiveMath (previously known as MathView)
Thinking Skills: Induction and Deduction
At the end of the lesson, the students should be able to:
(1) find the determinant of a matrix,
(2) find the inverse of a matrix,
(3) state the conditions for the inverse of a matrix to exist.

A. IDENTITY MATRIX

1. Let M = and I = .

Question 1: Calculate the products MI and IM. What do you notice?

2. The matrix I = is called the identity matrix. All the elements along the principal diagonal are ones; all the other elements are zeros. The product of any matrix with the identity matrix is itself.

3. Notice that all identity matrices are square matrices, e.g. the 3 x 3 identity matrix is .

B. DETERMINANT OF A MATRIX

1. You have learnt how to add, subtract and multiply matrices. Is it possible to divide matrices? In this lesson, you will learn how to find the inverse of a 2 x 2 matrix. The inverse is something like "division" but not exactly.

If you cannot view the LiveMath document below, click here to troubleshoot.

2. The 1st LiveMath document (IT3AMMatrices1.thp) shows the multiplication of two 2 x 2 square matrices M and N.

3. Change the values of a to h according to the matrices M and N in the table in the Worksheet (as shown below). Record the product MN and the value of ad–bc in the same table.

No. M N MN ad–bc

1.

2.

3.

4.

5.

Question 2: What is the pattern for the product MN? Express it in a general form, using k for the first element.

Question 3: Express the product MN in terms of the identity matrix I.

Question 4: What do you notice between the value of k and the value of ad–bc?

ad–bc is called the determinant of the matrix. It is denoted by det M, det , |M| or . You will learn about its use later on.

Question 5: Look at the elements of the 2 matrices M and N in the table above. If M = , express N in terms of a, b, c and d.

C. INVERSE OF A MATRIX

1. The inverse of a square matrix M, denoted by M^–1, is defined in such a way that MM^–1 = M^–1M = I, where I is the identity matrix.

2. In Q3, MN = kI. If k is not equal to 0, you can divide by k on both sides. Then you will get M(N) = I. Compare this with the definition of M^–1 in paragraph 1 and answer Q6.

Question 6: Express M^–1 in terms of N and k.

Question 7: Comparing Q4, Q5 and Q6, express M^–1 in terms of a, b, c and d, where M = .

3. For example, M = and N = .
Then MN = =
= –2
=
So M^–1 = = since MM^–1 = I.

4. Therefore = and = , i.e. M^–1M is also equal to I.

5. Using the first LiveMath document above, set e = d, f = –b, g = –c and h = a. Observe that the product MN will give kI whenever you change the values of a, b, c and d for the matrix M.

Question 8: By looking at the definition of the inverse in paragraph 1, i.e. MM^–1 = I, explain why the inverse of a matrix is something like "division" but not exactly.
Hint: Compare aa^–1 = 1 for arithmetic and algebra.

D. SINGULAR MATRIX

1. For determinants (and thus inverses) to exist, the matrices must be square matrices (or n x n matrices).

Question 9: Notice in Q3 that the last matrix M = has product MN = where the determinant k = 0. Is it possible to find the inverse of this matrix? Why?
Hint: Look at the answer for Q6.

Question 10: Hence summarise the condition for the inverse of a square matrix to exist.

Question 11: Suggest a reason why k = ad–bc is called the determinant.

2. A matrix whose determinant is zero is called a singular matrix because it does not have an inverse (no 'partner'). A non-singular matrix has a non-zero determinant and thus has an inverse.

E. EXERCISE

1. Do NOT use your computer to do the exercise.

2. Using the formulae for det M and M^–1 which you derived in Q4 and Q7, find and record the determinant and the inverse (if it exists) of the matrices M in the table in the Worksheet.



3.	Check your answers using the second LiveMath document (IT3AMMatrices2.thp) above.

F. ENRICHMENT

Why study matrices? Who invented them? Click here to find out more...

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Question 1:	Calculate the products MI and IM. What do you notice?


2.	The matrix I = is called the identity matrix. All the elements along the principal diagonal are ones; all the other elements are zeros. The product of any matrix with the identity matrix is itself.

3.	Notice that all identity matrices are square matrices, e.g. the 3 x 3 identity matrix is .


Question 2:	What is the pattern for the product MN? Express it in a general form, using k for the first element.

Question 3:	Express the product MN in terms of the identity matrix I.

Question 4:	What do you notice between the value of k and the value of ad–bc? ad–bc is called the determinant of the matrix. It is denoted by det M, det , \|M\| or . You will learn about its use later on.

Question 5:	Look at the elements of the 2 matrices M and N in the table above. If M = , express N in terms of a, b, c and d.

1.	The inverse of a square matrix M, denoted by M^–1, is defined in such a way that MM^–1 = M^–1M = I, where I is the identity matrix.

2.	In Q3, MN = kI. If k is not equal to 0, you can divide by k on both sides. Then you will get M(N) = I. Compare this with the definition of M^–1 in paragraph 1 and answer Q6.


Question 6:	Express M^–1 in terms of N and k.

Question 7:	Comparing Q4, Q5 and Q6, express M^–1 in terms of a, b, c and d, where M = .


3.	For example, M = and N = . Then MN = = = –2 = So M^–1 = = since MM^–1 = I.

4.	Therefore = and = , i.e. M^–1M is also equal to I.

5.	Using the first LiveMath document above, set e = d, f = –b, g = –c and h = a. Observe that the product MN will give kI whenever you change the values of a, b, c and d for the matrix M.


Question 8:	By looking at the definition of the inverse in paragraph 1, i.e. MM^–1 = I, explain why the inverse of a matrix is something like "division" but not exactly. Hint: Compare aa^–1 = 1 for arithmetic and algebra.


Question 9:	Notice in Q3 that the last matrix M = has product MN = where the determinant k = 0. Is it possible to find the inverse of this matrix? Why? Hint: Look at the answer for Q6.

Question 10:	Hence summarise the condition for the inverse of a square matrix to exist.

Question 11:	Suggest a reason why k = ad–bc is called the determinant.


2.	A matrix whose determinant is zero is called a singular matrix because it does not have an inverse (no 'partner'). A non-singular matrix has a non-zero determinant and thus has an inverse.

1.	Do NOT use your computer to do the exercise.

2.	Using the formulae for det M and M^–1 which you derived in Q4 and Q7, find and record the determinant and the inverse (if it exists) of the matrices M in the table in the Worksheet.