Matrix Addition, Subtraction, and Multiplication - Pre-Calculus
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What is $I_2\begin{bmatrix}a&b\c&d\end{bmatrix}$?
What is $I_2\begin{bmatrix}a&b\c&d\end{bmatrix}$?
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$\begin{bmatrix}a&b\c&d\end{bmatrix}$. Identity matrix preserves all entries when multiplied.
$\begin{bmatrix}a&b\c&d\end{bmatrix}$. Identity matrix preserves all entries when multiplied.
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What is $AB$ for $A=\begin{pmatrix}2&-1&0\end{pmatrix}$ and $B=\begin{pmatrix}3\4\-2\end{pmatrix}$?
What is $AB$ for $A=\begin{pmatrix}2&-1&0\end{pmatrix}$ and $B=\begin{pmatrix}3\4\-2\end{pmatrix}$?
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$2$. $(2)(3)+(-1)(4)+(0)(-2)=6-4+0=2$.
$2$. $(2)(3)+(-1)(4)+(0)(-2)=6-4+0=2$.
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What is the identity matrix property for multiplication with $I_n$?
What is the identity matrix property for multiplication with $I_n$?
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$I_nA=A$ and $AI_n=A$ (when dimensions match). Identity matrix acts as multiplicative identity element.
$I_nA=A$ and $AI_n=A$ (when dimensions match). Identity matrix acts as multiplicative identity element.
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What is the dot-product formula for an entry $(AB)_{ij}$?
What is the dot-product formula for an entry $(AB)_{ij}$?
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$(AB){ij}=\sum{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot product with column $j$ of $B$.
$(AB){ij}=\sum{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot product with column $j$ of $B$.
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If $A$ is $m\times n$ and $B$ is $n\times p$, what is the size of $AB$?
If $A$ is $m\times n$ and $B$ is $n\times p$, what is the size of $AB$?
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$AB$ is $m\times p$. Product has rows of $A$ and columns of $B$.
$AB$ is $m\times p$. Product has rows of $A$ and columns of $B$.
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What condition on dimensions must be true to multiply matrices $AB$?
What condition on dimensions must be true to multiply matrices $AB$?
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If $A$ is $m\times n$, then $B$ must be $n\times p$. Number of columns in $A$ must equal number of rows in $B$.
If $A$ is $m\times n$, then $B$ must be $n\times p$. Number of columns in $A$ must equal number of rows in $B$.
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What is the entrywise rule for matrix subtraction $(A-B)_{ij}$?
What is the entrywise rule for matrix subtraction $(A-B)_{ij}$?
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$(A-B){ij}=a{ij}-b_{ij}$. Subtract corresponding entries at the same position in both matrices.
$(A-B){ij}=a{ij}-b_{ij}$. Subtract corresponding entries at the same position in both matrices.
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What is the entrywise rule for matrix addition $(A+B)_{ij}$?
What is the entrywise rule for matrix addition $(A+B)_{ij}$?
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$(A+B){ij}=a{ij}+b_{ij}$. Add corresponding entries at the same position in both matrices.
$(A+B){ij}=a{ij}+b_{ij}$. Add corresponding entries at the same position in both matrices.
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What condition on dimensions must be true to subtract matrices $A-B$?
What condition on dimensions must be true to subtract matrices $A-B$?
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$A$ and $B$ must have the same dimensions $m \times n$. Subtraction requires matching rows and columns to subtract corresponding entries.
$A$ and $B$ must have the same dimensions $m \times n$. Subtraction requires matching rows and columns to subtract corresponding entries.
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What condition on dimensions must be true to add matrices $A$ and $B$?
What condition on dimensions must be true to add matrices $A$ and $B$?
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$A$ and $B$ must have the same dimensions $m \times n$. Addition requires matching rows and columns to add corresponding entries.
$A$ and $B$ must have the same dimensions $m \times n$. Addition requires matching rows and columns to add corresponding entries.
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What is $A-B$ for $A=\begin{pmatrix}7&1\-3&4\end{pmatrix}$ and $B=\begin{pmatrix}2&-5\6&0\end{pmatrix}$?
What is $A-B$ for $A=\begin{pmatrix}7&1\-3&4\end{pmatrix}$ and $B=\begin{pmatrix}2&-5\6&0\end{pmatrix}$?
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$\begin{pmatrix}5&6\-9&4\end{pmatrix}$. Subtract corresponding entries: $(7-2, 1-(-5); -3-6, 4-0)$.
$\begin{pmatrix}5&6\-9&4\end{pmatrix}$. Subtract corresponding entries: $(7-2, 1-(-5); -3-6, 4-0)$.
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What is $AB$ for $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$ and $B=\begin{pmatrix}0&1\-1&2\end{pmatrix}$?
What is $AB$ for $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$ and $B=\begin{pmatrix}0&1\-1&2\end{pmatrix}$?
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$\begin{pmatrix}-2&5\-4&11\end{pmatrix}$. Row 1: $(1)(0)+(2)(-1)=-2$, $(1)(1)+(2)(2)=5$; Row 2: $(3)(0)+(4)(-1)=-4$, $(3)(1)+(4)(2)=11$.
$\begin{pmatrix}-2&5\-4&11\end{pmatrix}$. Row 1: $(1)(0)+(2)(-1)=-2$, $(1)(1)+(2)(2)=5$; Row 2: $(3)(0)+(4)(-1)=-4$, $(3)(1)+(4)(2)=11$.
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What is the size (dimensions) of $AB$ if $A$ is $m\times n$ and $B$ is $n\times p$?
What is the size (dimensions) of $AB$ if $A$ is $m\times n$ and $B$ is $n\times p$?
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$AB$ is $m\times p$. Product has $A$'s rows and $B$'s columns.
$AB$ is $m\times p$. Product has $A$'s rows and $B$'s columns.
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What is the formula for an entry of a product matrix $(AB)_{ij}$?
What is the formula for an entry of a product matrix $(AB)_{ij}$?
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$(AB){ij}=\sum{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ of $A$ with column $j$ of $B$.
$(AB){ij}=\sum{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ of $A$ with column $j$ of $B$.
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What is $AB$ if $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}0&1\-1&2\end{bmatrix}$?
What is $AB$ if $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}0&1\-1&2\end{bmatrix}$?
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$\begin{bmatrix}-2&5\-4&11\end{bmatrix}$. Row-column products: $(1,2)\cdot(0,-1)=-2$, $(1,2)\cdot(1,2)=5$, etc.
$\begin{bmatrix}-2&5\-4&11\end{bmatrix}$. Row-column products: $(1,2)\cdot(0,-1)=-2$, $(1,2)\cdot(1,2)=5$, etc.
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Which statement is true about commutativity of matrix multiplication: $AB=BA$ always?
Which statement is true about commutativity of matrix multiplication: $AB=BA$ always?
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False; in general $AB\ne BA$ (and one product may be undefined). Matrix multiplication is not commutative.
False; in general $AB\ne BA$ (and one product may be undefined). Matrix multiplication is not commutative.
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What is $AB$ if $A=\begin{bmatrix}1\-2\3\end{bmatrix}$ and $B=\begin{bmatrix}4&0\end{bmatrix}$?
What is $AB$ if $A=\begin{bmatrix}1\-2\3\end{bmatrix}$ and $B=\begin{bmatrix}4&0\end{bmatrix}$?
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$\begin{bmatrix}4&0\-8&0\12&0\end{bmatrix}$. Column times row gives outer product matrix.
$\begin{bmatrix}4&0\-8&0\12&0\end{bmatrix}$. Column times row gives outer product matrix.
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What is the multiplicative identity for an $n\times n$ matrix $A$?
What is the multiplicative identity for an $n\times n$ matrix $A$?
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The identity matrix $I_n$, so $AI_n=I_nA=A$. Identity matrix leaves any square matrix unchanged.
The identity matrix $I_n$, so $AI_n=I_nA=A$. Identity matrix leaves any square matrix unchanged.
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What is the scalar multiplication rule for entries of $cA$?
What is the scalar multiplication rule for entries of $cA$?
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$(cA){ij}=c,a{ij}$. Multiply each entry by the scalar $c$.
$(cA){ij}=c,a{ij}$. Multiply each entry by the scalar $c$.
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What is $3A$ if $A=\begin{bmatrix}-1&2\0&5\end{bmatrix}$?
What is $3A$ if $A=\begin{bmatrix}-1&2\0&5\end{bmatrix}$?
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$\begin{bmatrix}-3&6\0&15\end{bmatrix}$. Multiply each entry by 3: $3(-1)=-3$, $3(2)=6$, $3(0)=0$, $3(5)=15$.
$\begin{bmatrix}-3&6\0&15\end{bmatrix}$. Multiply each entry by 3: $3(-1)=-3$, $3(2)=6$, $3(0)=0$, $3(5)=15$.
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What distributive property relates multiplication over addition: $A(B+C)$?
What distributive property relates multiplication over addition: $A(B+C)$?
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$A(B+C)=AB+AC$ (when dimensions are compatible). Matrix multiplication distributes over addition.
$A(B+C)=AB+AC$ (when dimensions are compatible). Matrix multiplication distributes over addition.
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What is $A+B$ if $A=\begin{bmatrix}1&-2\3&0\end{bmatrix}$ and $B=\begin{bmatrix}4&5\-1&2\end{bmatrix}$?
What is $A+B$ if $A=\begin{bmatrix}1&-2\3&0\end{bmatrix}$ and $B=\begin{bmatrix}4&5\-1&2\end{bmatrix}$?
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$\begin{bmatrix}5&3\2&2\end{bmatrix}$. Add corresponding entries: $1+4=5$, $-2+5=3$, $3+(-1)=2$, $0+2=2$.
$\begin{bmatrix}5&3\2&2\end{bmatrix}$. Add corresponding entries: $1+4=5$, $-2+5=3$, $3+(-1)=2$, $0+2=2$.
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What is $A-B$ if $A=\begin{bmatrix}6&1\-2&4\end{bmatrix}$ and $B=\begin{bmatrix}3&7\5&-1\end{bmatrix}$?
What is $A-B$ if $A=\begin{bmatrix}6&1\-2&4\end{bmatrix}$ and $B=\begin{bmatrix}3&7\5&-1\end{bmatrix}$?
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$\begin{bmatrix}3&-6\-7&5\end{bmatrix}$. Subtract corresponding entries: $6-3=3$, $1-7=-6$, $-2-5=-7$, $4-(-1)=5$.
$\begin{bmatrix}3&-6\-7&5\end{bmatrix}$. Subtract corresponding entries: $6-3=3$, $1-7=-6$, $-2-5=-7$, $4-(-1)=5$.
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What is $A+(-A)$ for any matrix $A$?
What is $A+(-A)$ for any matrix $A$?
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The zero matrix of the same size as $A$. A matrix plus its negative gives all zero entries.
The zero matrix of the same size as $A$. A matrix plus its negative gives all zero entries.
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What is the additive identity matrix for addition with an $m\times n$ matrix $A$?
What is the additive identity matrix for addition with an $m\times n$ matrix $A$?
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The $m\times n$ zero matrix $0$, so $A+0=A$. Adding zero matrix leaves any matrix unchanged.
The $m\times n$ zero matrix $0$, so $A+0=A$. Adding zero matrix leaves any matrix unchanged.
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What is $AB$ if $A=\begin{bmatrix}2&-1&0\end{bmatrix}$ and $B=\begin{bmatrix}3\4\5\end{bmatrix}$?
What is $AB$ if $A=\begin{bmatrix}2&-1&0\end{bmatrix}$ and $B=\begin{bmatrix}3\4\5\end{bmatrix}$?
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$2$. Row vector times column vector gives scalar: $2(3)+(-1)(4)+0(5)=2$.
$2$. Row vector times column vector gives scalar: $2(3)+(-1)(4)+0(5)=2$.
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What condition on dimensions is required to add matrices $A$ and $B$?
What condition on dimensions is required to add matrices $A$ and $B$?
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$A$ and $B$ must have the same dimensions ($m\times n$). Addition requires corresponding entries to exist.
$A$ and $B$ must have the same dimensions ($m\times n$). Addition requires corresponding entries to exist.
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What is $(AB)C$ compared to $A(BC)$ when all products are defined?
What is $(AB)C$ compared to $A(BC)$ when all products are defined?
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$(AB)C=A(BC)$. Matrix multiplication is associative.
$(AB)C=A(BC)$. Matrix multiplication is associative.
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What is the required dimension condition for the product $AB$ to be defined?
What is the required dimension condition for the product $AB$ to be defined?
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If $A$ is $m\times n$ and $B$ is $n\times p$. Number of columns in $A$ must equal number of rows in $B$.
If $A$ is $m\times n$ and $B$ is $n\times p$. Number of columns in $A$ must equal number of rows in $B$.
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What is $AB$ for $A=\begin{pmatrix}1&0\-2&3\end{pmatrix}$ and $B=\begin{pmatrix}5\-1\end{pmatrix}$?
What is $AB$ for $A=\begin{pmatrix}1&0\-2&3\end{pmatrix}$ and $B=\begin{pmatrix}5\-1\end{pmatrix}$?
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$\begin{pmatrix}5\-13\end{pmatrix}$. Row 1: $(1)(5)+(0)(-1)=5$; Row 2: $(-2)(5)+(3)(-1)=-10-3=-13$.
$\begin{pmatrix}5\-13\end{pmatrix}$. Row 1: $(1)(5)+(0)(-1)=5$; Row 2: $(-2)(5)+(3)(-1)=-10-3=-13$.
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