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Matrix Multiplication

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Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

4 Activities

Wize University Linear Algebra Textbook > Matrices

Elementary Matrices

5 Activities

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
e.g. Express the following matrix A‾\underline{A} as a product of five (or fewer) elementary matrices); repeat for  ⁣A‾−1 ⁣\!\underline{A}^{−1}\!,
More Matrix Inverse Questions:
Properties of Matrix Inverse
Suppose A,BA,B are any 2 n×nn\times n matrices, and that A+BA+B is invertible.
Show that A(A+B)−1B=B(A+B)−1AA(A+B)^{-1}B=B(A+B)^{-1}A .
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} }} then the (3,2)\bcb{\boldsymbol{ (3, 2)}} entry in adj(A)\bcb{\boldsymbol{ \text{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
Transpose of a Matrix
Prove that, for an invertible matrix A‾\bcb{\A} , A‾T\bcb{\mtran{\A}} is also invertible, and (AT)−1=(A−1)T\bcb{(A^T )^ {−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}.
a) find the inverse of 𝐴.
b) solve the system of linear equations 2𝑥+𝑦+3𝑧=32𝑥+𝑦+3𝑧=3
Practice Question: Matrix Properties
Practice Question: Matrix Properties
Let AA and BB be invertible matrices. Which of the following statements is/are always true?
i.) AB=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] is invertible, what is the third row of AA−1AA^{-1}?
Equation problem
Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)and AAis invertible, which of the following is an expression for A−1A^{-1}?
Invertible Matrix
Let AAand BBbe square matrices and UUis an invertible matrix. Given that BU=ABU=A, which of the following statements is always true?
Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)and AAis invertible, which of the following is an expression for A−1A^{-1}?
Let AAand BBbe square matrices and UUis an invertible matrix. Given that BU=ABU=A, which of the following statements is always true?
Let AAand BBbe n×nn\times nmatrices, and let kkbe 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} and BBis a 3×23\times2 matrix. If the product AB=[34011−2]AB = \begin{bmatrix}3 & 4\\0 & 1\\1 & -2\end{bmatrix}, find the (1,2)-entry of BB(i.e. B12B_{12}).
Practice: Matrix Inverse Application
Practice: Matrix Inverse Application
Let A=[120−1]A=\left[\begin{array}{c} 1&2\\0&-1 \end{array}\right] and suppose that 3AB−[1−120]=I23AB- \left[\begin{array}{c} 1&-1\\ 2&0 \end{array}\right] =I_2.
Find the matrix BB.
Practice: Matrix Properties (~2015 Test2 #16)
Practice: Matrix Properties
Let AA and BB be invertible matrices. Which of the following statements is/are always true? (select all that apply)
i.) AB=BAAB=BA
Inverse of a Matrix
e.g. Prove that, for an invertible matrix and a constant cc, cA‾c\underline{ A} is also invertible, and  ⁣(cA‾)−1=c−1A‾−1 ⁣\!(c \underline{A})^{ −1} = c^{ −1}\underline{A}^{−1}\!.
Transpose of a Matrix
e.g. Prove that, for an invertible matrix A,ATA, A^T is also invertible, and (AT)−1=(A−1)T(A^T )^ {−1} = (A^{−1} )^ T
Let AAand BBbe n×nn\times nmatrices, and let kkbe 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}, find the (3,3)-entry of A−1A^{-1}if 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] is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c} 1&p&0\\ -3&q&4\\ 2&r&1 \end{array}\right], find the value of rr.
Practice: Matrix Inverse
Let AA, BB, CC and DD 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:
Matrix Multiplication
Express the following matrix A‾\bcb{ \A } as a product of five (or fewer) elementary matrices); repeat for A‾−1\bcb{ \minv{\A} }.
A‾=[−300201010]\bcb{ \A=\begin{bmatrix} -3&0&0\\2&0&1\\0&1&0 \end{bmatrix} }
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] is invertible, what is the third row of AA−1AA^{-1}?
Practice: Matrix Inverse
If A=[abcdefghi]A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} is an invertible matrix and C=AA−1C=AA^{-1}, find the first row of CC .
Practice: Matrix Inverse
Let AA, BB, CC and DD be invertible matrices of size 10×1010\times10
Select the expression which is equivalent to the matrix:
(A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T
Practice: Matrix Inverse
Practice Question: Matrix Inverse
Let A=[1312]A=\begin{bmatrix} 1&3\\ 1&2 \end{bmatrix}. Find all 2×22\times2 matrices BB such that AB=[54−20].AB=\begin {bmatrix}5&4\\-2&0\end{bmatrix}.
Practice: Matrix Operations
Let AA, BB, CC and DD be invertible matrices of size 10×1010\times 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:
Invertible Matrix Theorem
Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏} where A5×5A_{5 \times 5} is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])?
More Elementary Matrices Questions:
Write down matrix A−1A^{-1} as the product of elementary matrices if:
A=[01211021−1]A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
Write down matrix A−1A^{-1} as the product of elementary matrices if:
A=[01211021−1]A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
Elementary matrices
Which of the following are elementary matrices?
A=[10001000−3]B=[0001001001001000]A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
C=[001010100]D=[10−71]C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
Example: Elementary Matrix
Find an elementary matrix EE such that A=EBA=EB if:
A=[246−1−1−11−13]A=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 1&-1&3 \end{array}\right] and B=[246−1−1−1315]B=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 3&1&5 \end{array}\right]
Which of the following are elementary matrices?
A=[10001000−3]B=[0001001001001000]A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
C=[001010100]D=[10−71]C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
Practice Question: Matrix of an ERO
Let EE be the elementary matrix obtained from I2I_2 by multiplying R2 by 1/3. Let BB be any 2×22\times2 matrix. Show that EBEB is the result of multiplying R2 of BB by 1/3. Find the inverse of EE and show that it is an elementary matrix.
Matrix Multiplication
Express the following matrix A‾\bcb{ \A } as a product of five (or fewer) elementary matrices); repeat for A‾−1\bcb{ \minv{\A} }.
A‾=[−300201010]\bcb{ \A=\begin{bmatrix} -3&0&0\\2&0&1\\0&1&0 \end{bmatrix} }
Practice: Elementary Matrix
Find an elementary matrix EE such that EB=CEB=C if:
B=[11300364]B=\left[\begin{array}{rr} 1&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right] and C=[101300364]C=\left[\begin{array}{rr} 10&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right]