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5.2: Special Types of Matrices

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Diagonal Matrices

We have already investigated, in exercises in the previous section, one special type of matrix. That was the zero matrix, and found that it behaves in matrix algebra in an analogous fashion to the real number 0; that is, as the additive identity. We will now investigate the properties of a few other special matrices.

Definition 5.2.1: Diagonal Matrix

A square matrix D is called a diagonal matrix if dij = 0 whenever ij.

Example 5.2.1: Some Diagonal Matrices

A=(100020005), B=(300000005), and I=(100010001) are all diagonal matrices.

Identity Matrix and Matrix Inverses

In the example above, the 3×3 diagonal matrix I whose diagonal entries are all 1's has the distinctive property that for any other 3×3 matrix A we have AI=IA=A. For example:

Example 5.2.2: Multiplying by the Identity Matrix

If A=(125672330), then AI=(125672330) and IA=(125672330).

In other words, the matrix I behaves in matrix algebra like the real number 1; that is, as a multiplicative identity. In matrix algebra, the matrix I is called simply the identity matrix. Convince yourself that if A is any n×n matrix AI=IA=A.

Definition 5.2.2: Identity Matrix

The n×n diagonal matrix In whose diagonal components are all 1's is called the identity matrix. If the context is clear, we simply use I.

In the set of real numbers we recall that, given a nonzero real number x, there exists a real number y such that xy=yx=1. We know that real numbers commute under multiplication so that the two equations can be summarized as xy=1. Further we know that y=x1=1x. Do we have an analogous situation in Mn×n(R)? Can we define the multiplicative inverse of an n×n matrix A? It seems natural to imitate the definition of multiplicative inverse in the real numbers.

Definition 5.2.3: Matrix Inverse

Let A be an n×n matrix. If there exists an n×n matrix B such that AB=BA=I, then B is a multiplicative inverse of A (called simply an inverse of A) and is denoted by A1

When we are doing computations involving matrices, it would be helpful to know that when we find A1, the answer we obtain is the only inverse of the given matrix. This would let us refer to the inverse of a matrix. We refrained from saying that in the definition, but the theorem below justifies it.

Remark: Those unfamiliar with the laws of matrix algebra should return to the following proof after they have familiarized themselves with the Laws of Matrix Algebra in Section 5.5.

Theorem 5.2.1: Inverses are Unique

The inverse of an n×n matrix A, when it exists, is unique.

Proof

Let A be an n×n matrix. Assume to the contrary, that A has two (different) inverses, say B and C. Then

B=BI Identity property of I=B(AC) Assumption that C is an inverse of A=(BA)C Associativity of matrix multiplication=IC Assumption that B is an inverse of A=C Identity property of I

Let A=(2003) . What is A1 ? Without too much difficulty, by trial and error, we determine that A1=(120013) . This might lead us to guess that the inverse is found by taking the reciprocal of all nonzero entries of a matrix. Alas, it isn't that easy!

If A=(1235) , the “reciprocal rule” would tell us that the inverse of A is B=(1121315). Try computing AB and you will see that you don't get the identity matrix. So, what is A1? In order to understand more completely the notion of the inverse of a matrix, it would be beneficial to have a formula that would enable us to compute the inverse of at least a 2×2 matrix. To do this, we introduce the definition of the determinant of a 2×2 matrix.

Definition 5.2.4: Determinant of a 2 by 2 Matrix

Let A=(abcd). The determinant of A is the number detA=adbc.

In addition to detA, common notation for the determinant of matrix A is |A|. This is particularly common when writing out the whole matrix, which case we would write |abcd| for the determinant of the general 2×2 matrix.

Example 5.2.3: Some Determinants of Two by Two Matrices

If A=(1235) then detA=152(3)=11. If B=(1224) then detB=1422=0.

Theorem 5.2.2: Inverse of 2 by 2 Matrix

Let A=(abcd). If detA0, then A1=1detA(dbca).

Proof

See Exercise 5.2.4 at the end of this section.

Example 5.2.4: Finding Inverses

Can we find the inverses of the matrices in Example 5.2.3? If A=(1235) then

A1=111(5231)=(511211311111)

The reader should verify that AA1=A1A=I.

The second matrix, B, has a determinant equal to zero. If we tried to apply the formula in Theorem 5.2.2, we would be dividing by zero. For this reason, the formula can't be applied and in fact B1 does not exist.

Remarks:

  • In general, if A is a 2×2 matrix and if detA=0, then A1 does not exist.
  • A formula for the inverse of n×n matrices n3 can be derived that also involves detA. Hence, in general, if the determinant of a matrix is zero, the matrix does not have an inverse. However the formula for even a 3×3 matrix is very long and is not the most efficient way to compute the inverse of a matrix.
  • In Chapter 12 we will develop a technique to compute the inverse of a higher-order matrix, if it exists.
  • Matrix inversion comes first in the hierarchy of matrix operations; therefore, AB1 is A(B1).

Exercises

Exercise 5.2.1

For the given matrices A find A1 if it exists and verify that AA1=A1A=I. If A1 does not exist explain why.

  1. A=(1321)
  2. A=(6384)
  3. A=(1301)
  4. A=(1001)
  5. Use the definition of the inverse of a matrix to find A1: A=(3000120005)
Answer
  1. (1/53/52/51/5)
  2. No inverse exists.
  3. (1301)
  4. A1=A
  5. (1/300020001/5)

Exercise 5.2.2

For the given matrices A find A1 if it exists and verify that AA1=A1A=I. If A1 does not exist explain why.

  1. A=(2112)
  2. A=(0102)
  3. A=(1c01)
  4. A=(abba), where |a||b|.

Exercise 5.2.3

  1. Let A=(2314) and B=(3321). Verify that (AB)1=B1A1.
  2. Let A and B be n×n invertible matrices. Prove that (AB)1=B1A1. Why is the right side of the above statement written “backwards”? Is this necessary? Hint: Use Theorem 5.2.1
Answer

Let A and B be n by n invertible matrices.

(B1A1)(AB)=(B1)(A1(AB))=(B1)((A1A)B)=((B1)IB)=B1(B)=I

Similarly, (AB)(B1A1)=I.

By Theorem 5.2.1, B1A1 is the only inverse of AB. If we tried to invert AB with A1B1, we would be unsuccessful since we cannot rearrange the order of the matrices.

Exercise 5.2.4

Let A=(abcd). Derive the formula for A1.

Exercise 5.2.5: Linearity of Determinants

  1. Let A and B be 2-by-2 matrices. Show that det(AB)=(detA)(detB).
  2. It can be shown that the statement in part (a) is true for all n×n matrices. Let A be any invertible n×n matrix. Prove that det(A1)=(detA)1. Note: The determinant of the identity matrix In is 1 for all n.
  3. Verify that the equation in part (b) is true for the matrix in Exercise 5.2.1(a) of this section.
Answer

1=detI=det(AA1)=detA detA1. Now solve for detA1.

Exercise 5.2.6

Prove by induction that for n1, (a00b)n=(an00bn).

Exercise 5.2.7

Use the assumptions in Exercise 5.2.5 to prove by induction that if n1, det(An)=(detA)n.

Answer

Basis: (n=1):detA1=detA=(detA)1

Induction: Assume detAn=(detA)n for some n1.

detAn+1=det(AnA) by the definition of exponents=det(An)det(A) by exercise 5=(detA)n(detA) by the induction hypothesis =(detA)n+1

Exercise 5.2.8

Prove: If the determinant of a matrix A is zero, then A does not have an inverse. Hint: Use the indirect method of proof and exercise 5.2.5.

Exercise 5.2.9

  1. Let A,B, and D be n×n matrices. Assume that B is invertible. If A=BDB1 , prove by induction that Am=BDmB1 is true for m1.
  2. Given that A=(815611)=B(1002)B1 where B=(5332) what is A10?
Answer
  1. Assume A=BDB1
    Basis: (m=1): A1=A=BD1B1 is given.
    Induction: Assume that for some positive integer m, Am=BDmB1
    Am+1=AmA=(BDmB1)(BDB1)by the induction hypothesis=(BDm(B1B)(DB1)by associativity=BDmDB1by the definition of inverse=BDm+1B1
  2. A10=BD10B1=(920615345613810231)

This page titled 5.2: Special Types of Matrices is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Al Doerr & Ken Levasseur via source content that was edited to the style and standards of the LibreTexts platform.

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