Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

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Matrix Inverse
A square matrix AAA is said to be invertible (or non-singular) if there exists a matrix A−1A^{-1}A−1 (called the inverse of AAA) such that:
 AA−1=A−1A=IAA^{-1}=A^{-1}A=IAA−1=A−1A=I
Example
If A=[13−1−2]A=
\left[
\begin{array}{rr}
1&3\\
-1&-2
\end{array}
\right]A=[1−1​3−2​], show that A−1=[−2−311]A^{-1}=
\left[
\begin{array}{rr}
-2&-3\\
1&1
\end{array}
\right]A−1=[−21​−31​].

AA−1=[13−1−2][−2−311]=[1001]=I✓AA^{-1}=
\left[
\begin{array}{rr}
1&3\\
-1&-2
\end{array}
\right]
\left[
\begin{array}{rr}
-2&-3\\
1&1
\end{array}
\right]
=
\left[
\begin{array}{rr}
1&0\\
0&1
\end{array}
\right]
=I
\quad
\colorThree{\checkmark}AA−1=[1−1​3−2​][−21​−31​]=[10​01​]=I✓
A−1A=[−2−311][13−1−2]=[1001]=I✓A^{-1}A=
\left[
\begin{array}{rr}
-2&-3\\
1&1
\end{array}
\right]

\left[
\begin{array}{rr}
1&3\\
-1&-2
\end{array}
\right]

=
\left[
\begin{array}{rr}
1&0\\
0&1
\end{array}
\right]
=I
\quad
\colorThree{\checkmark}A−1A=[−21​−31​][1−1​3−2​]=[10​01​]=I✓

PAGE BREAK
Analogy With Real Numbers
Consider the real number product 18(8) = 8−1(8) = 1\dfrac{1}{8}(8) \ =\ 8^{-1}(8) \ =\ 181​(8) = 8−1(8) = 1.
8−18^{-1}8−1 is the multiplicative inverse of 888: it is the unique number that multiplies 888 to get a product of 111.
Recall that III is the matrix equivalent of the number 111 for multiplication (since AI=IA=AAI=IA=AAI=IA=A).
Just like with real numbers, if we want to "cancel out" a matrix with multiplication, we multiply by its inverse: A−1A=IA^{-1}A = IA−1A=I.
PAGE BREAK
Properties of the Matrix Inverse
Let A, BA,\ BA, B be invertible matrices and let k∈Rk \in \realsk∈R.
(A−1)−1=A(A^{-1})^{-1}=A(A−1)−1=A
(kA)−1=1kA−1(kA)^{-1}=\frac{1}{k}A^{-1}(kA)−1=k1​A−1
(AT)−1=(A−1)T(A^T)^{-1}=(A^{-1})^T(AT)−1=(A−1)T
(AB)−1=B−1A−1(AB)^{-1}=B^{-1}A^{-1}(AB)−1=B−1A−1
A−1A^{-1}A−1 is unique

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Example: Matrix Inverse
Given the following matrices AAA and BBB:
A=[−11−12],  B=[−21−11]A=\begin{bmatrix}-1 & 1\\-1 & 2\end{bmatrix},\ \ B=\begin{bmatrix}-2 & 1\\-1 & 1\end{bmatrix}A=[−1−1​12​],  B=[−2−1​11​]
A) Determine whether BBB is the inverse of AAA.

Multiply the matrices together, and if the product is the identity, then they are inverse:
AB=[−11−12][−21−11]=[1001]=IAB=\begin{bmatrix}-1 & 1\\-1 & 2\end{bmatrix}\begin{bmatrix}-2 & 1\\-1 & 1\end{bmatrix}=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}=IAB=[−1−1​12​][−2−1​11​]=[10​01​]=I
BA=[−21−11][−11−12]=[1001]=IBA=\begin{bmatrix}-2 & 1\\-1 & 1\end{bmatrix}\begin{bmatrix}-1 & 1\\-1 & 2\end{bmatrix}=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}=IBA=[−2−1​11​][−1−1​12​]=[10​01​]=I
So BBB is indeed the inverse of AAA (B=A−1B=A^{-1}B=A−1).

B) Is AAA the inverse of BBB?

Yes, AAA is also the inverse of BBB:
Since B=A−1B=A^{-1}B=A−1, it follows that B−1=(A−1)−1=AB^{-1}=(A^{-1})^{-1} = AB−1=(A−1)−1=A.

If A=[abcdefghi]A=
\begin{bmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{bmatrix}A=​adg​beh​cfi​​ is an invertible matrix and C=AA−1C=AA^{-1}C=AA−1, find the first row of CCC .

[abc]\begin{bmatrix}
  a & b & c
\end{bmatrix}[a​b​c​]

[111]\begin{bmatrix}
  1 & 1 & 1
\end{bmatrix}[1​1​1​]

[100]\begin{bmatrix}
  1 & 0 & 0
\end{bmatrix}
[1​0​0​]

[000]\begin{bmatrix}
  0 & 0 & 0
\end{bmatrix}[0​0​0​]

[a−c  d−f  g−i]\begin{bmatrix}
  a-c\ \  & d-f\ \  & g-i
\end{bmatrix}[a−c  ​d−f  ​g−i​]

I don't know

If [1ax2by0cz]\left[\begin{array}{c}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

r=

I don't know

Extra Practice

Properties of Matrix Inverse

Suppose A,BA,BA,B are any 2 n×nn\times nn×n matrices, and that A+BA+BA+B is invertible.
Show that A(A+B)−1B=B(A+B)−1AA(A+B)^{-1}B=B(A+B)^{-1}AA(A+B)−1B=B(A+B)−1A .

133 - FML 3 - 18.1W e.g. 75

   ⁣A‾=  [123−201−1−23]\bcb{\boldsymbol{ \ul{A} =  \begin{bmatrix} 1 & 2 & 3 \\ -2 & 0 & 1 \\ -1 & -2 & 3  \end{bmatrix} }}A​=  ​1−2−1​20−2​313​​ then the (3,2)\bcb{\boldsymbol{ (3, 2)}}(3,2) entry in adj(A)\bcb{\boldsymbol{ \text{adj}(A)}}adj(A) is:
(a) −4(b)  0(c)  4(d)  None of the above\text{(a)}\, \boldsymbol{  -4}
 \qquad\qquad\qquad
\text{(b)}\; \boldsymbol{  0}
 \qquad\qquad\qquad
\text{(c)} \;\boldsymbol{  4}
 \qquad\qquad\qquad
\text{(d)} \;\text{\bf{None of the above}}
 \qquad\qquad\qquad(a)−4(b)0(c)4(d)None of the above

Transpose of a Matrix

Prove that, for an invertible matrix A‾\bcb{\A}A​ , A‾T\bcb{\mtran{\A}}A​T is also invertible, and (AT)−1=(A−1)T\bcb{(A^T )^ {−1} = (A^{−1} )^ T}(AT)−1=(A−1)T.

Given the matrix 𝐴=[2130101−12].\text{Given the matrix 𝐴}=\begin{bmatrix}
2&1&3\\
0&1&0\\
1&-1&2\\
\end{bmatrix}.Given the matrix A=​201​11−1​302​​.
a) find the inverse of 𝐴. 
b) solve the system of linear equations          2𝑥+𝑦+3𝑧=32𝑥+𝑦+3𝑧=32x+y+3z=3                                                                  

Practice Question: Matrix Properties

Practice Question: Matrix Properties
Let AAA and BBB be invertible matrices. Which of the following statements is/are always true?
i.) AB=BAAB=BAAB=BA

Practice Question: Matrix Inverse

Practice Question: Matrix Inverse
If A=[abcxyz10−1]A=\left[\begin{array}{l}
a&b&c\\
x&y&z\\
1&0&-1
\end{array}\right]A=​ax1​by0​cz−1​​ is invertible, what is the third row of AA−1AA^{-1}AA−1?

Equation problem

Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)A2=(I−2A)(I+A)and AAAis invertible, which of the following is an expression for A−1A^{-1}A−1?

Invertible Matrix

Let AAAand BBBbe square matrices and UUUis an invertible matrix. Given that BU=ABU=ABU=A, which of the following statements is always true?

Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)A2=(I−2A)(I+A)and AAAis invertible, which of the following is an expression for A−1A^{-1}A−1?

Let AAAand BBBbe square matrices and UUUis an invertible matrix. Given that BU=ABU=ABU=A, which of the following statements is always true?

Let AAAand BBBbe n×nn\times nn×nmatrices, and let kkkbe a constant. Which of the following statements is always true?

Suppose that A=[1−103−20041]A=\begin{bmatrix}1 & -1 & 0\\3 & -2 & 0\\0 & 4 & 1\end{bmatrix}A=​130​−1−24​001​​ and BBBis a 3×23\times23×2 matrix. If the product AB=[34011−2]AB = \begin{bmatrix}3 & 4\\0 & 1\\1 & -2\end{bmatrix}AB=​301​41−2​​, find the (1,2)-entry of BBB(i.e. B12B_{12}B12​).

Practice: Matrix Inverse Application

Practice: Matrix Inverse Application
Let A=[120−1]A=\left[\begin{array}{c}
1&2\\0&-1
\end{array}\right]A=[10​2−1​] and suppose that 3AB−[1−120]=I23AB-

\left[\begin{array}{c}
1&-1\\
2&0
\end{array}\right]

=I_23AB−[12​−10​]=I2​. 
Find the matrix BBB.

Practice: Matrix Properties (~2015 Test2 #16)

Practice: Matrix Properties
Let AAA and BBB be invertible matrices. Which of the following statements is/are always true? (select all that apply)
i.) AB=BAAB=BAAB=BA

Inverse of a Matrix

e.g. Prove that, for an invertible matrix  and a constant ccc,  cA‾c\underline{ A}cA​ is also invertible, and  ⁣(cA‾)−1=c−1A‾−1 ⁣\!(c \underline{A})^{ −1} = c^{ −1}\underline{A}^{−1}\!(cA​)−1=c−1A​−1.

Transpose of a Matrix

e.g. Prove that, for an invertible matrix A,ATA, A^TA,AT is also invertible, and (AT)−1=(A−1)T(A^T )^ {−1} = (A^{−1} )^ T(AT)−1=(A−1)T 

Let AAAand BBBbe n×nn\times nn×nmatrices, and let kkkbe a constant. Select the statement that is always true.

Given thatA=[3−32−2461−10]A=\begin{bmatrix}3 & -3 & 2\\-2 & 4 & 6\\1 & -1 & 0\end{bmatrix}A=​3−21​−34−1​260​​, find the (3,3)-entry of A−1A^{-1}A−1if it exists. 

Practice: Matrix Inverse (~2016 Test2 #17)

Practice: Matrix Inverse
If [1ax2by0cz]\left[\begin{array}{c}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

Practice: Matrix Inverse

Let AAA, BBB, CCC and DDD be size 10 by 10 invertible matrices: Select the expression which is equal to the matrix (A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T(A(BC)−1D)T:

Matrix Multiplication

Express the following matrix A‾\bcb{ \A } A​ as a product of five (or fewer) elementary matrices); repeat for A‾−1\bcb{ \minv{\A} }A​−1.
A‾=[−300201010]\bcb{
\A=\begin{bmatrix}
-3&0&0\\2&0&1\\0&1&0
\end{bmatrix}
}A​=​−320​001​010​​

Practice: Matrix Inverse (~2015 Test2 #17)

Practice: Matrix Inverse
If A=[abcxyz10−1]A=\left[\begin{array}{l}
a&b&c\\
x&y&z\\
1&0&-1
\end{array}\right]A=​ax1​by0​cz−1​​ is invertible, what is the third row of AA−1AA^{-1}AA−1?

Practice: Matrix Inverse

If A=[abcdefghi]A=
\begin{bmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{bmatrix}A=​adg​beh​cfi​​ is an invertible matrix and C=AA−1C=AA^{-1}C=AA−1, find the first row of CCC .

Practice: Matrix Inverse

Let AAA, BBB, CCC and DDD be invertible matrices of size 10×1010\times1010×10
Select the expression which is equivalent to the matrix:
(A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T(A(BC)−1D)T

Practice: Matrix Inverse

Practice Question: Matrix Inverse
Let A=[1312]A=\begin{bmatrix}
1&3\\
1&2
\end{bmatrix}A=[11​32​]. Find all 2×22\times22×2 matrices BBB such that AB=[54−20].AB=\begin
{bmatrix}5&4\\-2&0\end{bmatrix}.AB=[5−2​40​].

Practice: Matrix Operations

Let AAA, BBB, CCC and DDD be invertible matrices of size 10×1010\times 1010×10
Show that: (A(BC)−1D)T=DT(BT)−1(CT)−1AT\left(A\left(BC\right)^{-1}D\right)^T=D^T\left(B^{T}\right)^{-1}\left(C^{T}\right)^{-1}A^T(A(BC)−1D)T=DT(BT)−1(CT)−1AT:

Matrix Multiplication

e.g. Express the following matrix A‾\underline{A}A​ as a product of five (or fewer) elementary matrices); repeat for  ⁣A‾−1 ⁣\!\underline{A}^{−1}\!A​−1,

Invertible Matrix Theorem

Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏}Ax⃗=b where A5×5A_{5 \times 5}A5×5​ is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])rank([A ∣ b])?

Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

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Matrix Inverse

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Extra Practice

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133 - FML 3 - 18.1W e.g. 75

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