High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Linear Algebra Textbook > Matrices

Matrix Multiplication

Basics of MatricesPrevious Section
Matrix TransposeNext Section
Popular Courses
Find My Course
0:00 / 0:00

Matrix Multiplication

To multiply two matrices, their dimensions must be compatible.

The product ABAB exists only if the number of columns in matrix AA is equal to the number of rows in matrix BB.

If matrix AA is of size m×p\colorOne{m}\times \bm{\colorTwo{p}}, and matrix BB is of size p×n\bm{\colorTwo{p}} \times \colorThree{n}, the product ABAB will be of size: m×n\colorOne{m} \times \colorThree{n}
Am×p Bp×n=(AB)m×nA_{\colorOne{m} \times \bm{\colorTwo{p}}} \ B_{\bm{\colorTwo{p}} \times \colorThree{n}} = (AB)_{\colorOne{m} \times \colorThree{n}}

Wize Tip
Think of the "inner dimensions" as matching and cancelling.
If they don't match, the matrices cannot be multiplied (incompatible)!

PAGE BREAK

Multiplying Matrices

The (i,j)(i,j)-entry of the product ABAB is the dot product of row ii of matrix AA with column jj of matrix BB:
[AB]ij=[ai1ai2⋯aip][b1jb2j⋮bpj]=ai1b1j+ai2b2j+⋯+aipbpj[AB]_{ij}= \left[ \begin{array}{cccc} a_{i1} & a_{i2} & \cdots & a_{ip} \end{array} \right] \left[ \begin{array}{c} b_{1j} \\[0.2em] b_{2j} \\[0.2em] \vdots \\[0.2em] b_{pj} \end{array} \right] = a_{i1}b_{1j} + a_{i2}b_{2j}+\cdots+a_{ip}b_{pj}

Example
Consider the rows of the first matrix and the columns of the second matrix.
Take the dot product of each of these pairs of rows and columns:

[124503][3−152]=[1(3)+2(5)1(−1)+2(2)4(3)+5(5)4(−1)+5(2)0(3)+3(5)0(−1)+3(2)]=[133376156]\left[\begin{array}{rr} \colorThree{1} & \colorThree{2}\\[0.5em] {4} & {5}\\[0.5em] \colorFive{0} & \colorFive{3} \end{array}\right] \left[\begin{array}{rr} \colorTwo{3} & \colorOne{-1}\\[0.5em] \colorTwo{5} & \colorOne{2} \end{array} \right] = \left[\begin{array}{rr} \colorThree{1}\colorTwo{(3)} + \colorThree{2}\colorTwo{(5)} & \quad \colorThree{1}\colorOne{(-1)} + \colorThree{2}\colorOne{(2)}\\[0.5em] {4}\colorTwo{(3)} + {5}\colorTwo{(5)} & \quad {4}\colorOne{(-1)} + {5}\colorOne{(2)}\\[0.5em] \colorFive{0}\colorTwo{(3)} + \colorFive{3}\colorTwo{(5)} & \quad \colorFive{0}\colorOne{(-1)} + \colorFive{3}\colorOne{(2)}\\ \end{array}\right] = \left[ \begin{array}{rrr} 13 & 3\\[0.5em] 37 & 6\\[0.5em] 15 & 6\\ \end{array} \right]


PAGE BREAK

Properties of Matrix Multiplication

Watch Out!
In general: AB≠BAAB\neq BA
The order in which we write the matrices matters!
  • A(B+C)=AB+ACA(B+C)=AB+AC
  • (A+B)C=AC+BC(A+B)C=AC+BC
  • A(BC)=(AB)CA(BC)=(AB)C
  • AI=IA=AAI = IA = A
Note: If AB=0AB=0, it does not necessarily mean that AA or BB is the zero matrix!
PAGE BREAK

Matrix Powers

Just like with real numbers, we can multiply a square matrix by itself, raising it to the power of some integer kk.
Ak=AA⋯A⏟k timesA^k = \underbrace{AA\cdots A}_{k\text{ times}}

Properties

  • A0=IA^0=I
  • (cA)k=ckAk(cA)^k=c^kA^k
Watch Out!
(AB)k≠AkBk(AB)^k \textcolor{red}{\neq}A^kB^k

Since matrix multiplication is not commutative, powers do not distribute over matrix multiplication:

(AB)k=(AB)(AB)⋯(AB)⏟k times(AB)^k = \underbrace{(AB)(AB)\cdots(AB)}_{k\text{ times}}


0:00 / 0:00

Example: Matrix Multiplication Criteria

Given that AA is 3×23\times2, BB is 2×22\times2, CC is 4×34\times3, and DD is 5×25\times2, which of the following operations are defined?
A) ABAB
Well-defined. The result is a matrix of size (3×2)⏞A(2×2)⏞B=3×2\overbrace{(3\times \cancel2)}^A \overbrace{(\cancel2\times 2)}^B = 3\times 2 .
B) BABA
Not defined since the inner dimensions do not match: (2×2)⏞A(3×2)⏞B\overbrace{(2\times \colorTwo{2})}^A \overbrace{(\colorTwo{3}\times 2)}^B .
C) DB2DB^2
Well-defined. The result is a matrix of size (5×2)⏞D(2×2)⏞B(2×2)⏞B=5×2\overbrace{(5\times \cancel{\colorThree{2}})}^D \overbrace{(\cancel{\colorThree{2}} \times \cancel{\colorOne{2}})}^B \overbrace{(\cancel{\colorOne{2}} \times 2)}^B = 5\times 2.
0:00 / 0:00

Example: Matrix Multiplication

Compute the products ABAB and BABA given the following matrices:
A=[135−2−4−6]A=\left[\begin{array}{rrr}1&3&5\\-2&-4&-6\end{array}\right], B=[2−11201]B=\left[\begin{array}{rr}2 & -1\\1&2\\0&1\end{array}\right]
Note that AA is 2×32\times 3 and BB is 3×23\times 2, so the product ABAB is well-defined and is of size (2×3)⏞A(3×2)⏞B=2×2\overbrace{(2\times \cancel3)}^A \overbrace{(\cancel3\times 2)}^B = 2\times 2.
AB=[135−2−4−6][2−11201]=[1(2)+3(1)+5(0)1(−1)+3(2)+5(1)−2(2)−4(1)−6(0)−2(−1)−4(2)−6(1)]=[510−8−12]\begin{array}{rcl} AB&=&\left[\begin{array}{rrr}1&3&5\\-2&-4&-6\end{array}\right]\left[\begin{array}{rr}2 & -1\\1&2\\0&1\end{array}\right]\\[2em] &=&\left[\begin{array}{cc} 1(2)+3(1)+5(0) & 1(-1)+3(2)+5(1)\\[0.5em] -2(2)-4(1)-6(0)&-2(-1)-4(2)-6(1) \end{array}\right]\\[2em] &=&\left[\begin{array}{rr} 5 & 10\\ -8&-12 \end{array}\right]\\[2em] \end{array}
The product BABA is also well-defined, and will be of size (3×2)⏞B(2×3)⏞A=3×3\overbrace{(3\times \cancel2)}^B \overbrace{(\cancel2\times 3)}^A = 3\times 3.
BA=[2−11201][135−2−4−6]=[2(1)−1(−2)2(3)−1(−4)2(5)−1(−6)1(1)+2(−2)1(3)+2(−4)1(5)+2(−6)0(1)+1(−2)0(3)+1(−4)0(5)+1(−6)]=[41016−3−5−7−2−4−6]\begin{array}{rcl} BA&=&\left[\begin{array}{rr}2 & -1\\1&2\\0&1\end{array}\right]\left[\begin{array}{rrr}1&3&5\\-2&-4&-6\end{array}\right]\\[2.5em] &=& \left[\begin{array}{cc} 2(1)-1(-2) & \quad2(3)-1(-4) & \quad2(5)-1(-6)\\[0.5em] 1(1)+2(-2) & \quad1(3)+2(-4) & \quad1(5)+2(-6)\\[0.5em] 0(1)+1(-2) & \quad0(3)+1(-4) & \quad0(5)+1(-6)\\ \end{array}\right]\\[3em] &=&\left[\begin{array}{rr} 4 & 10 & 16\\[0.5em] -3 & -5 & -7\\[0.5em] -2 & -4 & -6 \end{array}\right]\\[2em] \end{array}

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]
Compute MNMN.
Given the matrix A=[1011]A=\begin{bmatrix} 1&0\\1&1 \end{bmatrix}, find AnA^n for any integer n≥1n \ge 1.

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]
Compute ABAB.

Extra Practice
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25
Show that A‾T  ⁣A‾ \bcb{\boldsymbol{ \mtran{\A}\ul{A}}} \, is a symmetric matrix for any n×n\bcb{\boldsymbol{ n \times n}} matrix   ⁣A‾\bcb{\boldsymbol{ \ul{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 = \vb can be thought of as taking the dot product of the rows of the matrix A‾\A with the column vector x⃗\vx. If A⃗1j\vec{A}_{1j} represents the first row of the matrix A‾\A, 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}
133 - FML 3 - 18.1W e.g. 74
Find   ⁣A‾7\bcb{\boldsymbol{ \ul{A}^7}} if   ⁣A‾=[2420]\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 2 & 4 \\ 2 & 0 \end{bmatrix} }}.
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} }}, find A‾⋅ı^\bcb{\boldsymbol{ \underline{A} \cdot \ihat}}. Repeat for A‾⋅ȷ^\bcb{\boldsymbol{ \underline{A}\cdot \jhat}}. What do you notice?
A‾⋅ı^= \A \cdot \ihat = \,
a
c
A‾⋅ȷ^= \A \cdot \jhat = \,
b
d
If A=[21−25]A=\begin{bmatrix}2&1\\-2&5\end{bmatrix} and B=[4−332]B=\begin{bmatrix}4&-3\\3&2\end{bmatrix}, find (1,2)-entry of (2A−1(3B)T)−1\left(2A^{-1}\left(3B\right)^T\right)^{-1}
If A=[21−25]A = \begin{bmatrix} 2 & 1 \\ -2 & 5 \end{bmatrix} and B=[4−332]B = \begin{bmatrix} 4 & -3 \\ 3 & 2 \end{bmatrix}, find the (1,2) entry of (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}, find AnA^n for integer values n>1n>1
Matrix Multiplication
Practice: Matrix Multiplication
Compute MNMN and NMNM, 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]
Matrix Transpose
Given C=[431]C = \left[ \begin{array}{rrr} 4 & 3 & 1 \end{array} \right] and D=[131245]D = \left[ \begin{array}{rrr} 1 & 3 & 1\\ 2 & 4 & 5\\ \end{array} \right], 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}, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, if D=ACD=AC, find D23D_{23} . (i.e. find the (2, 3)-entry of DD).
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]
Practice: Matrix Powers
Practice: Matrix Powers
Given the matrix A=[1−11−1]A=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}, find AnA^n, where nn 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]
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}, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, if D=ACD=AC, find D23D_{23} . (i.e. find the (2, 3)-entry of DD).
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]
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}, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, which of the following is/are defined?
i.) (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}, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, 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}, find the third column of (B−2I3)2(B-2I_3)^2.
Matrix Properties With Powers
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}
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+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})
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}, B=[1−2]B=\begin{bmatrix}1 & -2\end{bmatrix} and C=[02]C=\begin{bmatrix}0\\2\end{bmatrix}
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}, B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, find the second row of (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] and N=[−1−301]N=\left[ \begin{array}{r} -1\\[0.5em] -3\\[0.5em] 0\\[0.5em] 1 \end{array} \right]
Compute the products MNMN and NMNM
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] and B=[2−1−301−1003]B=\left[ \begin{array}{rrr} 2&-1&-3\\ 0&1&-1\\ 0&0&3 \end{array} \right]
Compute the products ABAB, BABA, and A2A^2
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})
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}and B=[152−15130.38−16.78] B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix}. Find the (3,2)-entry of ABAB(i.e. AB32AB_{32}).
Matrix Operations
Suppose that AAand BBare both 3×33\times3matrices. Find a matrix BBsuch that BABAresults in swapping row 1 and row 2 in AA, followed by multiplying row 3 in AAby 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}, B=[1−2]B=\begin{bmatrix}1 & -2\end{bmatrix} and C=[02]C=\begin{bmatrix}0\\2\end{bmatrix}
Suppose that AAand BBare both 3×33\times3matrices. Find a matrix BBsuch that BABAresults in swapping row 1 and row 2 in AA, followed by multiplying row 3 in AAby 4.
If AA is a 2×42\times4matrix, BBis a 2×22\times2matrix, and CCis a 4×24\times2matrix, 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} and B=[152−15130.38−16.78] B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix} Find the (3,2)-entry of ABAB(i.e. AB32AB_{32}).
133 - FML 1 - 18.1W e.g. 8
If B ⁣‾ =diag(β1,β2)\B=\text{diag}(\beta_1,\beta_2) find B ⁣‾ 2\B^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], find A21A_{21} (i.e. the (2,1)-entry of the matrix A).
Practice: Matrix Properties (~2014 Test2 #13)
Practice: Matrix Properties
Let AA andBB be n×nn\times n matrices such that AB=OAB=O, where matrix OO is the n×nn\times n zero matrix.
Which of the following statement must always be true for any nn value?
Given the two matrices below, find ABAB and BABA .
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}
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:
Practice Question: Matrix Multiplication
Practice Question: Matrix Multiplication
Given the matrices B=[115400]B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}, and C=[00001−4]C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}, find the second row of (BC)T(BC)^T.
Matrix Multiplication
e.g. If A‾=[−810−3−7] \A=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix} and B ⁣‾ =[−2046] ⁣\B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!, find ABA B
Matrix Multiplication
If A‾=[−810−3−7]\bcb{ {\A}=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix}} and B ⁣‾ =[−2046] ⁣\bcb{ \B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!}, find A‾B ⁣‾ \bcb{\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}
Let A=[1312]A=\begin{bmatrix}1 & 3\\1&2\end{bmatrix}. Find all 2×22\times 2 matrices BB such that AB=0AB=0.
Matrix Multiplication
Prove or disprove the following statement: if B‾\underline{B} has a column of zeros, then so does  ⁣A‾ B‾ ⁣\!\underline{A}\ \underline{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], and B=[−1−273]B=\left[\begin{array}{rr}-1&-2\\7&3\end{array}\right], compute 3AB3AB
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 and Identity
Let A=[−1010−22010]A=\left[\begin{array}{rrr} -1&0&1\\0&-2&2\\0&1&0 \end{array}\right], find (A+I)2(A+I)^2
Matrix Multiplication
Suppose that A=[10−12]A=\begin{bmatrix} 1&0\\-1&2 \end{bmatrix} and B=[3−1−20]B=\begin{bmatrix} 3&-1\\-2&0 \end{bmatrix}. Find AB2AB^2.
Matrix Multiplication
Suppose that A=[2−11103]A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix} and B=[03−2−110]B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}.
Find B3B^3.
Matrix Multiplication and Transpose
Suppose that A=[2−11103]A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix} and B=[03−2−110]B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}.
Find 2ATB−(BTA)T−ATIB2A^TB-(B^TA)^T-A^TIB, where II is the 3x3 identity matrix.