Matrix Properties With Powers

Matrix Multiplication

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For each matrix A, find A3 if it exists


a)A=[2222]  b) A=[222222]a)   A=\begin{bmatrix}2&2\\2&2\\\end{bmatrix} \space \space
b) \space A=\begin{bmatrix}2&2&2\\2&2&2\\\end{bmatrix}a)A=[22​22​]  b) A=[22​22​22​]

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$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25

Show that A‾T  ⁣A‾ \bcb{\boldsymbol{ \mtran{\A}\ul{A}}} \,A​TA​  is a symmetric matrix for any n×n\bcb{\boldsymbol{ n \times n}}n×n matrix   ⁣A‾\bcb{\boldsymbol{ \ul{A} }}A​. 

19.4F_WML_6_$\tkco{eg3}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{3}$_

Recall that matrix-vector multiplication of the form A‾x⃗=b⃗\A \vx = \vbA​x=b can be thought of as taking the dot product of the rows of the matrix A‾\AA​ with the column vector x⃗\vxx. If A⃗1j\vec{A}_{1j}A1j​ represents the first row of the matrix A‾\AA​, write the following matrix-vector multiplication as a system of vector-vector products:
[a11a12a13a21a22a23a31a32a33][x1x2x3]=[b1b2b3]
          \begin{bmatrix}
          a_{11} & a_{12} & a_{13} \\
          a_{21} & a_{22} & a_{23} \\
          a_{31} & a_{32} & a_{33} \\
          \end{bmatrix}
          %
          \colvecth{x_1}{x_2}{x_3}
          %
          = 
          %
          \colvecth{b_1}{b_2}{b_3}
        ​a11​a21​a31​​a12​a22​a32​​a13​a23​a33​​​​x1​x2​x3​​​=​b1​b2​b3​​​

133 - FML 3 - 18.1W e.g. 74

 Find   ⁣A‾7\bcb{\boldsymbol{ \ul{A}^7}}A​7 if   ⁣A‾=[2420]\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 2 & 4  \\ 2 & 0  \end{bmatrix} }}A​=[22​40​].

133 - FML 3 - 18.1W e.g. 5

Given the matrix A‾=[abcd]\bcb{\boldsymbol{ \underline{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} }}A​=[ac​bd​], find A‾⋅ı^\bcb{\boldsymbol{ \underline{A} \cdot \ihat}}A​⋅^. Repeat for A‾⋅ȷ^\bcb{\boldsymbol{ \underline{A}\cdot \jhat}}A​⋅^​. What do you notice?
A‾⋅ı^= \A \cdot \ihat = \, A​⋅^= a

              c

A‾⋅ȷ^= \A \cdot \jhat = \, A​⋅^​= b

              d

If A=[21−25]A=\begin{bmatrix}2&1\\-2&5\end{bmatrix}A=[2−2​15​] and B=[4−332]B=\begin{bmatrix}4&-3\\3&2\end{bmatrix}B=[43​−32​], find (1,2)-entry of (2A−1(3B)T)−1\left(2A^{-1}\left(3B\right)^T\right)^{-1}(2A−1(3B)T)−1

If A=[21−25]A = \begin{bmatrix}
2 & 1 \\
-2 & 5
\end{bmatrix}A=[2−2​15​] and B=[4−332]B = \begin{bmatrix}
4 & -3 \\
3 & 2
\end{bmatrix}B=[43​−32​], find the (1,2) entry of (2A−1(3B)T)−1(2A^{-1}(3B)^T)^{-1}(2A−1(3B)T)−1

Practice: Matrix Powers (~2014 Test2 #14)

Practice: Matrix Powers
Given the matrix A=[1011]A=\begin{bmatrix}
1&0\\1&1
\end{bmatrix}A=[11​01​], find AnA^nAn for integer values n>1n>1n>1

Matrix Multiplication

Practice: Matrix Multiplication
Compute MNMNMN and NMNMNM, where
M=[6−103],N=[−1−301]M=\left[
\begin{array}{rrrr}
6&-1&0&3\end{array}
\right]
,\qquad
N=\left[
\begin{array}{r}
-1\\[0.5em]
-3\\[0.5em]
0\\[0.5em]
1
\end{array}
\right]M=[6​−1​0​3​],N=​−1−301​​

Matrix Transpose

Given C=[431]C
=
\left[
\begin{array}{rrr}
4 & 3 & 1
\end{array}
\right]C=[4​3​1​] and D=[131245]D
=
\left[
\begin{array}{rrr}
1 & 3 & 1\\
2 & 4 & 5\\
\end{array}
\right]D=[12​34​15​], find the following entries.

Practice Question 3: Matrix Multiplication

Practice Question: Matrix Multiplication
Given the matricesA=[1−1023−3]A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}A=​103​−12−3​​, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], if D=ACD=ACD=AC, find D23D_{23}D23​ . (i.e. find the (2, 3)-entry of DDD).

Matrix Multiplication

Practice: Matrix Multiplication
A=[−331−220−112],B=[2−1−301−1003]A=\left[
\begin{array}{rrr}
-3&3&1\\
-2&2&0\\
-1&1&2
\end{array}
\right]
,\qquad

B=\left[
\begin{array}{rrr}
2&-1&-3\\
0&1&-1\\
0&0&3
\end{array}
\right]A=​−3−2−1​321​102​​,B=​200​−110​−3−13​​

Practice: Matrix Powers

Practice: Matrix Powers
Given the matrix A=[1−11−1]A=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}A=[11​−1−1​], find AnA^nAn, where nnn is an integer greater than 1.

Practice: Matrix Operations

Given the matrices below, which of the following statements is/are defined?
A=[abcd], B=[efghijkl], C=[mnopqrst]A=\left[\begin{array}{c}
a&b\\c&d
\end{array}\right],
\ 
B=\left[\begin{array}{c}
e&f&g&h\\i&j&k&l
\end{array}\right],\ 
C=\left[\begin{array}{c}
m&n\\o&p\\q&r\\s&t
\end{array}\right]A=[ac​bd​], B=[ei​fj​gk​hl​], C=​moqs​nprt​​

Practice: Matrix Multiplication (~2016 Test2 #13)

Practice: Matrix Multiplication
Given the matricesA=[1−1023−3]A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}A=​103​−12−3​​, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], if D=ACD=ACD=AC, find D23D_{23}D23​ . (i.e. find the (2, 3)-entry of DDD).

Matrix Multiplication

Practice: Matrix Multiplication
M=[6−103],N=[−1−301]M=\left[
\begin{array}{rrrr}
6&-1&0&3\end{array}
\right]
,\qquad
N=\left[
\begin{array}{r}
-1\\[0.5em]
-3\\[0.5em]
0\\[0.5em]
1
\end{array}
\right]M=[6​−1​0​3​],N=​−1−301​​

Practice Question 1: Matrix Operations

Practice Question: Matrix Operations
Given the matrices A=[1−1023−3]A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}A=​103​−12−3​​, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], which of the following is/are defined?
i.) (BC)T(BC)^T(BC)T

Practice: Matrix Operations (~2014 Test2 #9)

Practice: Matrix Operations
Given the matrices A=[1−1023−3]A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}A=​103​−12−3​​, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], which of the following is/are defined? (select all that apply)

Practice: Matrix Operations (~2014 Test2 #12)

Practice: Matrix Operations
Given the matrix B=[201013002]B=\begin{bmatrix}
2&0&1\\0&1&3\\0&0&2
\end{bmatrix}B=​200​010​132​​, find the third column of (B−2I3)2(B-2I_3)^2(B−2I3​)2.

Practice Question: Matrix Multiplication Criteria

Practice Question: Properties of Matrices
𝐴 is a 2 × 5 matrix, 𝐵 is a 5 × 6 matrix, 𝐶 is a 6 × 2 matrix, and 𝐷 is a 1 × 2 row matrix. Which one of the following is defined?
a. AB+2CAB+2CAB+2C

Matrix Multiplication

Prove or disprove that for all square matrices of the same size:
A‾2−B‾2=(A‾+B‾)(A‾−B‾)\underline{A}^2-\underline{B}^2=(\underline{A}+\underline{B})(\underline{A}-\underline{B})A​2−B​2=(A​+B​)(A​−B​) 

Practice: Matrix Multiplication

Consider the matrices and vectors defined below. Which of the multiplications below are not well defined?
A=[245−5]A=\begin{bmatrix}2 & 4\\5 & -5\end{bmatrix}A=[25​4−5​], B=[1−2]B=\begin{bmatrix}1 & -2\end{bmatrix}B=[1​−2​] and C=[02]C=\begin{bmatrix}0\\2\end{bmatrix}C=[02​]

Practice: Matrix Multiplication (~2015 Test2 #13)

Practice: Matrix Multiplication
Given the matricesA=[1−1023−3]A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}A=​103​−12−3​​, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], find the second row of (BC)T(BC)^T(BC)T.

Practice: Matrix Multiplication of different sized matrices

Let M=[6−103]M=\left[
\begin{array}{rrrr}
6&-1&0&3\end{array}
\right]M=[6​−1​0​3​] and N=[−1−301]N=\left[
\begin{array}{r}
-1\\[0.5em]
-3\\[0.5em]
0\\[0.5em]
1
\end{array}
\right]N=​−1−301​​
Compute the products MNMNMN and NMNMNM

Practice: Matrix Multiplication of 3x3 matrices

Let A=[−331−220−112]A=\left[
\begin{array}{rrr}
-3&3&1\\
-2&2&0\\
-1&1&2
\end{array}
\right]A=​−3−2−1​321​102​​ and B=[2−1−301−1003]B=\left[
\begin{array}{rrr}
2&-1&-3\\
0&1&-1\\
0&0&3
\end{array}
\right]B=​200​−110​−3−13​​
Compute the products ABABAB, BABABA, and A2A^2A2

Matrix Multiplication

e.g. Prove or disprove that for all square matrices of the same size, 
A‾2−B‾2=(A‾+B‾)(A‾−B‾)\underline{A}^2-\underline{B}^2=(\underline{A}+\underline{B})(\underline{A}-\underline{B})A​2−B​2=(A​+B​)(A​−B​) 

Matrix problem

Suppose that  A=[1/520.88−1520.03111−10]A=\begin{bmatrix}1/5 & 2 & 0.88\\-152 & 0.03 & 11\\1 & -1 & 0\end{bmatrix}A=​1/5−1521​20.03−1​0.88110​​and B=[152−15130.38−16.78] B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix}B=​15−150.38​213−16.78​​.  Find the (3,2)-entry of ABABAB(i.e. AB32AB_{32}AB32​).

Matrix Operations

Suppose that AAAand BBBare both 3×33\times33×3matrices. Find a matrix BBBsuch that BABABAresults in swapping row 1 and row 2 in AAA, followed by multiplying row 3 in AAAby 4.

Matrix Multiplication

Consider the matrices and vectors defined below. Which of the multiplications below are not well defined?
A=[245−5]A=\begin{bmatrix}2 & 4\\5 & -5\end{bmatrix}A=[25​4−5​], B=[1−2]B=\begin{bmatrix}1 & -2\end{bmatrix}B=[1​−2​] and C=[02]C=\begin{bmatrix}0\\2\end{bmatrix}C=[02​]

Suppose that AAAand BBBare both 3×33\times33×3matrices. Find a matrix BBBsuch that BABABAresults in swapping row 1 and row 2 in AAA, followed by multiplying row 3 in AAAby 4.

If AAA is a 2×42\times42×4matrix, BBBis a 2×22\times22×2matrix, and CCCis a 4×24\times24×2matrix, which of the following expressions is NOT defined?

Suppose that  A=[1/520.88−1520.03111−10]A=\begin{bmatrix}1/5 & 2 & 0.88\\-152 & 0.03 & 11\\1 & -1 & 0\end{bmatrix}A=​1/5−1521​20.03−1​0.88110​​ and B=[152−15130.38−16.78] B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix}B=​15−150.38​213−16.78​​ Find the (3,2)-entry of ABABAB(i.e. AB32AB_{32}AB32​).

133 - FML 1 - 18.1W e.g. 8

If B ⁣‾ =diag(β1,β2)\B=\text{diag}(\beta_1,\beta_2)B​=diag(β1​,β2​) find B ⁣‾ 2\B^2B​2.

Practice: Matrix Multiplication

Given that A=[13−1021][−22001−143−1]A=\left[\begin{array}{c}
1&3&-1\\
0&2&1
\end{array}\right]
\left[\begin{array}{c}
-2&2&0\\
0&1&-1\\
4&3&-1
\end{array}\right]A=[10​32​−11​]​−204​213​0−1−1​​, find A21A_{21}A21​ (i.e. the (2,1)-entry of the matrix A).

Practice: Matrix Properties (~2014 Test2 #13)

Practice: Matrix Properties
Let AAA andBBB be n×nn\times nn×n matrices such that AB=OAB=OAB=O, where matrix OOO is the n×nn\times nn×n zero matrix.
Which of the following statement must always be true for any nnn value?

Given the two matrices below, find ABABAB and BABABA .
A=[4−47−6140−40],B=[5−4−3−15−43−15]A=\begin{bmatrix}4&-4&7\\-6&1&4\\0&-4&0\end{bmatrix}, \quad
B=\begin{bmatrix}5&-4&-3\\-1&5&-4\\3&-1&5\end{bmatrix}A=​4−60​−41−4​740​​,B=​5−13​−45−1​−3−45​​

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:

Practice Question: Matrix Multiplication

Practice Question: Matrix Multiplication
Given the matrices B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}B=​150​140​​, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}C=[00​01​0−4​], find the second row of (BC)T(BC)^T(BC)T.

Matrix Multiplication

e.g.  If  A‾=[−810−3−7] \A=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix}A​=[−8−3​10−7​] and  B ⁣‾ =[−2046] ⁣\B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!B​=[−24​06​], find  ABA BAB 

Matrix Multiplication

If  A‾=[−810−3−7]\bcb{ {\A}=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix}}A​=[−8−3​10−7​] and  B ⁣‾ =[−2046] ⁣\bcb{ \B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!}B​=[−24​06​], find  A‾B ⁣‾ \bcb{\A \B}A​B​ .

Example: Matrix Multiplication

Example:
Computer the following matrix multiplications (matrix products):
a.) [1−24][03−3]\begin{bmatrix}1 &−2 &4\end{bmatrix}\begin{bmatrix}0\\3\\-3\end{bmatrix}[1​−2​4​]​03−3​​

Let A=[1312]A=\begin{bmatrix}1 & 3\\1&2\end{bmatrix}A=[11​32​]. Find all 2×22\times 22×2 matrices BBB such that AB=0AB=0AB=0.

Matrix Multiplication

Prove or disprove the following statement: if B‾\underline{B}B​ has a column of zeros, then so does  ⁣A‾ B‾ ⁣\!\underline{A}\ \underline{B}\!A​ B​, if this product is defined.

Matrix and Scalar Multiplication

Let A=[230−1]A=\left[\begin{array}{rr}2&3\\0&-1\end{array}\right]A=[20​3−1​], and B=[−1−273]B=\left[\begin{array}{rr}-1&-2\\7&3\end{array}\right]B=[−17​−23​], compute 3AB3AB3AB

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 and Identity

Let A=[−1010−22010]A=\left[\begin{array}{rrr}
-1&0&1\\0&-2&2\\0&1&0
\end{array}\right]A=​−100​0−21​120​​, find (A+I)2(A+I)^2(A+I)2

Matrix Multiplication

Suppose that A=[10−12]A=\begin{bmatrix}
1&0\\-1&2
\end{bmatrix}A=[1−1​02​] and B=[3−1−20]B=\begin{bmatrix}
3&-1\\-2&0
\end{bmatrix}B=[3−2​−10​]. Find AB2AB^2AB2.

Matrix Multiplication

Suppose that A=[2−11103]A=\begin{bmatrix}
2&-1\\
1&1\\
0&3
\end{bmatrix}A=​210​−113​​ and B=[03−2−110]B=\begin{bmatrix}
0&3\\
-2&-1\\
1&0
\end{bmatrix}B=​0−21​3−10​​.
Find B3B^3B3.

Matrix Multiplication and Transpose

Suppose that A=[2−11103]A=\begin{bmatrix}
2&-1\\
1&1\\
0&3
\end{bmatrix}A=​210​−113​​ and B=[03−2−110]B=\begin{bmatrix}
0&3\\
-2&-1\\
1&0
\end{bmatrix}B=​0−21​3−10​​.
Find 2ATB−(BTA)T−ATIB2A^TB-(B^TA)^T-A^TIB2ATB−(BTA)T−ATIB, where III is the 3x3 identity matrix.

Matrix Properties With Powers

Related Topics

Matrix Multiplication

More Matrix Multiplication Questions:

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25

19.4F_WML_6_$\tkco{eg3}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{3}$_

133 - FML 3 - 18.1W e.g. 74

133 - FML 3 - 18.1W e.g. 5

Practice: Matrix Powers (~2014 Test2 #14)

Matrix Multiplication

Matrix Transpose

Practice Question 3: Matrix Multiplication

Matrix Multiplication

Practice: Matrix Powers

Practice: Matrix Operations

Practice: Matrix Multiplication (~2016 Test2 #13)

Matrix Multiplication

Practice Question 1: Matrix Operations

Practice: Matrix Operations (~2014 Test2 #9)

Practice: Matrix Operations (~2014 Test2 #12)

Practice Question: Matrix Multiplication Criteria

Matrix Multiplication

Practice: Matrix Multiplication

Practice: Matrix Multiplication (~2015 Test2 #13)

Practice: Matrix Multiplication of different sized matrices

Practice: Matrix Multiplication of 3x3 matrices

Matrix Multiplication

Matrix problem

Matrix Operations

Matrix Multiplication

133 - FML 1 - 18.1W e.g. 8

Practice: Matrix Multiplication

Practice: Matrix Properties (~2014 Test2 #13)

Practice: Matrix Inverse

Practice Question: Matrix Multiplication

Matrix Multiplication

Matrix Multiplication

Example: Matrix Multiplication

Matrix Multiplication

Matrix and Scalar Multiplication

Practice: Matrix Operations

Matrix Multiplication and Identity

Matrix Multiplication

Matrix Multiplication

Matrix Multiplication and Transpose