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Invertible Matrix Theorem

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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Rank of a Matrix

4 Activities

Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

4 Activities

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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 Rank of a Matrix Questions:
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 15
The linear system M‾⋅x⃗=0⃗\bcb{\boldsymbol{ \underline{M} \cdot \vx = \vz}} is known to have only the trivial solution. If M‾\bcb{\boldsymbol{ \underline{M}}} is a matrix with 8 rows and 7 columns, what is its rank?
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
Example: Rank of a Matrix
Example:
1. Find the ranks of the following matrices and state the number of solutions.
a.) [102100∣120]\left[\begin{array}{l} 1& 0 \\ 2& 1 \\ 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
A linear system of three equations in four unknowns has
The reduced row echelon form of the augmented matrix for a linear system is
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
a) What is the rank of matrix A
Equivalent Conditions to Full Rank
a) If AA is a 5×35\times 3 matrix of full rank, and BB is a 3×73\times 7 matrix of rank 2, what are the possible values of rank(AB)rank(AB) ?
b) If AA is a 3×53\times 5 matrix of full rank and BB is a 5×45\times 4 matrix of rank 3, what are the possible values of rank(AB)rank(AB) ?
Find the rank of the following matrix:
[525151021200]\begin{bmatrix}5&2&5&1\\5&1&0&2\\1&2&0&0\end{bmatrix}
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b]\left[A\left|b\right.\right]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAis 3, which of the following statements best describes the system?
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b][A|b]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AA is rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AA is 3, which of the following statements best describes the system?
Linear equations
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Matrix of a Projection
a) Find the matrix AA so that for x⃗∈R3\vec x \in \mathbb{R}^3 the transformation mapping x⃗→proj(−13,1,12)x⃗\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x is the same as the transformation mapping x⃗→Ax⃗\vec x\rightarrow A\vec x
b) What is rank(A)rank(A) ?
Rank of a Matrix
Determine the number of solutions for the system of linear equations Ax⃗=b⃗A\vec{x}=\vec{b} with 5 equations and 5 unknowns, given:
Rank of a Matrix
In the reduced row echelon form of the coefficient matrix AA that corresponds to a system of linear equations with 6 equations in 9 unknowns, the maximum number of leading 1s is
6
.
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a matrix
Let AA be a 6 x 4 matrix and let Ax=bA\bm{x}=\bm{b} be a system of linear equations. Which of the following statements must always be true? (select all that apply)
Rank of a Matrix
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a Matrix
How many solutions does the system of linear equations corresponding to the following augmented matrix have?
[1010−1101101000011−1000000]\left[ \begin{array}{rrrrr|r} 1&0&1&0&-1&1\\ 0&1&1&0&1&0\\ 0&0&0&1&1&-1\\ 0&0&0&0&0&0 \end{array} \right]
Rank of a Matrix
Which of the following is an augmented matrix corresponding to a system of linear equation that is consistent? (select all that apply)
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:
Matrix Multiplication
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}\!,