Scalar Multiplication of Matrices - Pre-Calculus
Card 1 of 30
State the distributive property: how are $(k+m)A$ and $kA+mA$ related?
State the distributive property: how are $(k+m)A$ and $kA+mA$ related?
Tap to reveal answer
$(k+m)A=kA+mA$. Scalar addition distributes over scalar multiplication.
$(k+m)A=kA+mA$. Scalar addition distributes over scalar multiplication.
← Didn't Know|Knew It →
State the distributive property: how are $k(A+B)$ and $kA+kB$ related?
State the distributive property: how are $k(A+B)$ and $kA+kB$ related?
Tap to reveal answer
$k(A+B)=kA+kB$. Scalar multiplication distributes over matrix addition.
$k(A+B)=kA+kB$. Scalar multiplication distributes over matrix addition.
← Didn't Know|Knew It →
Compute $-1\cdot\begin{bmatrix}-2&0\5&-7\end{bmatrix}$.
Compute $-1\cdot\begin{bmatrix}-2&0\5&-7\end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix}2&0\-5&7\end{bmatrix}$. Change signs: $-1(-2)=2$, $-1(0)=0$, $-1(5)=-5$, $-1(-7)=7$.
$\begin{bmatrix}2&0\-5&7\end{bmatrix}$. Change signs: $-1(-2)=2$, $-1(0)=0$, $-1(5)=-5$, $-1(-7)=7$.
← Didn't Know|Knew It →
What is $(-1)A$ for any matrix $A$?
What is $(-1)A$ for any matrix $A$?
Tap to reveal answer
$(-1)A=-A$ (negate every entry). Multiplying by $-1$ changes the sign of every entry.
$(-1)A=-A$ (negate every entry). Multiplying by $-1$ changes the sign of every entry.
← Didn't Know|Knew It →
What is $1\cdot A$ for any matrix $A$?
What is $1\cdot A$ for any matrix $A$?
Tap to reveal answer
$1A=A$. Multiplying by 1 leaves the matrix unchanged (identity property).
$1A=A$. Multiplying by 1 leaves the matrix unchanged (identity property).
← Didn't Know|Knew It →
What is $0\cdot\begin{bmatrix}2&7\-1&3\end{bmatrix}$?
What is $0\cdot\begin{bmatrix}2&7\-1&3\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}0&0\0&0\end{bmatrix}$. Zero times any matrix gives the zero matrix.
$\begin{bmatrix}0&0\0&0\end{bmatrix}$. Zero times any matrix gives the zero matrix.
← Didn't Know|Knew It →
What is $\frac{1}{2}\begin{bmatrix}8&-6\end{bmatrix}$?
What is $\frac{1}{2}\begin{bmatrix}8&-6\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}4&-3\end{bmatrix}$. Multiply by $\frac{1}{2}$: $\frac{1}{2}(8)=4$, $\frac{1}{2}(-6)=-3$.
$\begin{bmatrix}4&-3\end{bmatrix}$. Multiply by $\frac{1}{2}$: $\frac{1}{2}(8)=4$, $\frac{1}{2}(-6)=-3$.
← Didn't Know|Knew It →
What is $-2\begin{bmatrix}5\-3\1\end{bmatrix}$?
What is $-2\begin{bmatrix}5\-3\1\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}-10\6\-2\end{bmatrix}$. Multiply each entry by $-2$: $-2(5)=-10$, $-2(-3)=6$, $-2(1)=-2$.
$\begin{bmatrix}-10\6\-2\end{bmatrix}$. Multiply each entry by $-2$: $-2(5)=-10$, $-2(-3)=6$, $-2(1)=-2$.
← Didn't Know|Knew It →
What is the size of $kA$ if $A$ is an $m\times n$ matrix and $k$ is a scalar?
What is the size of $kA$ if $A$ is an $m\times n$ matrix and $k$ is a scalar?
Tap to reveal answer
$kA$ is still an $m\times n$ matrix. Scalar multiplication doesn't change matrix dimensions.
$kA$ is still an $m\times n$ matrix. Scalar multiplication doesn't change matrix dimensions.
← Didn't Know|Knew It →
State the rule for scalar multiplication of a matrix $A$ by a scalar $k$.
State the rule for scalar multiplication of a matrix $A$ by a scalar $k$.
Tap to reveal answer
$kA=[k a_{ij}]$ (multiply every entry $a_{ij}$ by $k$). Each element is multiplied by the scalar.
$kA=[k a_{ij}]$ (multiply every entry $a_{ij}$ by $k$). Each element is multiplied by the scalar.
← Didn't Know|Knew It →
Find and correct the error: $2\begin{bmatrix}1&3\2&4\end{bmatrix}=\begin{bmatrix}2&3\2&8\end{bmatrix}$.
Find and correct the error: $2\begin{bmatrix}1&3\2&4\end{bmatrix}=\begin{bmatrix}2&3\2&8\end{bmatrix}$.
Tap to reveal answer
Correct: $2\begin{bmatrix}1&3\2&4\end{bmatrix}=\begin{bmatrix}2&6\4&8\end{bmatrix}$. Error: didn't multiply all entries; $2(3)=6$ not 3.
Correct: $2\begin{bmatrix}1&3\2&4\end{bmatrix}=\begin{bmatrix}2&6\4&8\end{bmatrix}$. Error: didn't multiply all entries; $2(3)=6$ not 3.
← Didn't Know|Knew It →
Compute $4\left(\frac{1}{2}A\right)$ for $A=\begin{bmatrix}6&-2\end{bmatrix}$.
Compute $4\left(\frac{1}{2}A\right)$ for $A=\begin{bmatrix}6&-2\end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix}12&-4\end{bmatrix}$. First: $\frac{1}{2}A=\begin{bmatrix}3&-1\end{bmatrix}$, then multiply by 4.
$\begin{bmatrix}12&-4\end{bmatrix}$. First: $\frac{1}{2}A=\begin{bmatrix}3&-1\end{bmatrix}$, then multiply by 4.
← Didn't Know|Knew It →
Compute $(3+(-1))A$ for $A=\begin{bmatrix}2&-4\1&0\end{bmatrix}$.
Compute $(3+(-1))A$ for $A=\begin{bmatrix}2&-4\1&0\end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix}4&-8\2&0\end{bmatrix}$. $(3+(-1))=2$, so multiply each entry by 2.
$\begin{bmatrix}4&-8\2&0\end{bmatrix}$. $(3+(-1))=2$, so multiply each entry by 2.
← Didn't Know|Knew It →
Compute the doubled payoff matrix of $P=\begin{bmatrix}-1&4\0&2\end{bmatrix}$.
Compute the doubled payoff matrix of $P=\begin{bmatrix}-1&4\0&2\end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix}-2&8\0&4\end{bmatrix}$. Multiply each payoff by 2: $2(-1)=-2$, $2(4)=8$, $2(0)=0$, $2(2)=4$.
$\begin{bmatrix}-2&8\0&4\end{bmatrix}$. Multiply each payoff by 2: $2(-1)=-2$, $2(4)=8$, $2(0)=0$, $2(2)=4$.
← Didn't Know|Knew It →
What happens to every payoff in a payoff matrix $P$ when you double the game’s payoffs?
What happens to every payoff in a payoff matrix $P$ when you double the game’s payoffs?
Tap to reveal answer
The new matrix is $2P$. Doubling payoffs means multiplying the payoff matrix by 2.
The new matrix is $2P$. Doubling payoffs means multiplying the payoff matrix by 2.
← Didn't Know|Knew It →
Find the missing entry $x$ if $-3\begin{bmatrix}2&x\end{bmatrix}=\begin{bmatrix}-6&9\end{bmatrix}$.
Find the missing entry $x$ if $-3\begin{bmatrix}2&x\end{bmatrix}=\begin{bmatrix}-6&9\end{bmatrix}$.
Tap to reveal answer
$x=-3$. Since $-3(x)=9$, divide: $x=9/(-3)=-3$.
$x=-3$. Since $-3(x)=9$, divide: $x=9/(-3)=-3$.
← Didn't Know|Knew It →
State the associative property for scalars: how are $k(mA)$ and $(km)A$ related?
State the associative property for scalars: how are $k(mA)$ and $(km)A$ related?
Tap to reveal answer
$k(mA)=(km)A$. Scalar multiplication is associative.
$k(mA)=(km)A$. Scalar multiplication is associative.
← Didn't Know|Knew It →
Identify the scalar $k$ if $k\begin{bmatrix}2&-1\end{bmatrix}=\begin{bmatrix}6&-3\end{bmatrix}$.
Identify the scalar $k$ if $k\begin{bmatrix}2&-1\end{bmatrix}=\begin{bmatrix}6&-3\end{bmatrix}$.
Tap to reveal answer
$k=3$. Since $k(2)=6$ and $k(-1)=-3$, we have $k=3$.
$k=3$. Since $k(2)=6$ and $k(-1)=-3$, we have $k=3$.
← Didn't Know|Knew It →
What is $2(A+B)$ if $A=\begin{bmatrix}1&0\2&-1\end{bmatrix}$ and $B=\begin{bmatrix}3&4\-2&5\end{bmatrix}$?
What is $2(A+B)$ if $A=\begin{bmatrix}1&0\2&-1\end{bmatrix}$ and $B=\begin{bmatrix}3&4\-2&5\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}8&8\0&8\end{bmatrix}$. First add matrices: $A+B=\begin{bmatrix}4&4\0&4\end{bmatrix}$, then multiply by $2$.
$\begin{bmatrix}8&8\0&8\end{bmatrix}$. First add matrices: $A+B=\begin{bmatrix}4&4\0&4\end{bmatrix}$, then multiply by $2$.
← Didn't Know|Knew It →
What is $(k+m)A$ if $A=\begin{bmatrix}1&-2\0&3\end{bmatrix}$, $k=2$, and $m=-5$?
What is $(k+m)A$ if $A=\begin{bmatrix}1&-2\0&3\end{bmatrix}$, $k=2$, and $m=-5$?
Tap to reveal answer
$\begin{bmatrix}-3&6\0&-9\end{bmatrix}$. Since $k+m=2+(-5)=-3$, multiply each entry by $-3$.
$\begin{bmatrix}-3&6\0&-9\end{bmatrix}$. Since $k+m=2+(-5)=-3$, multiply each entry by $-3$.
← Didn't Know|Knew It →
A payoff matrix is $P=\begin{bmatrix}1&-2\3&0\end{bmatrix}$. What is the doubled payoff matrix $2P$?
A payoff matrix is $P=\begin{bmatrix}1&-2\3&0\end{bmatrix}$. What is the doubled payoff matrix $2P$?
Tap to reveal answer
$\begin{bmatrix}2&-4\6&0\end{bmatrix}$. Double each payoff by multiplying each entry by $2$.
$\begin{bmatrix}2&-4\6&0\end{bmatrix}$. Double each payoff by multiplying each entry by $2$.
← Didn't Know|Knew It →
What is $-\frac{3}{2}\begin{bmatrix}2&-4\-6&8\end{bmatrix}$?
What is $-\frac{3}{2}\begin{bmatrix}2&-4\-6&8\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}-3&6\9&-12\end{bmatrix}$. Multiply each entry by $-\frac{3}{2}$.
$\begin{bmatrix}-3&6\9&-12\end{bmatrix}$. Multiply each entry by $-\frac{3}{2}$.
← Didn't Know|Knew It →
What is $-2\begin{bmatrix}1&5&0\-3&2&7\end{bmatrix}$?
What is $-2\begin{bmatrix}1&5&0\-3&2&7\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}-2&-10&0\6&-4&-14\end{bmatrix}$. Multiply each entry by $-2$ to get the result.
$\begin{bmatrix}-2&-10&0\6&-4&-14\end{bmatrix}$. Multiply each entry by $-2$ to get the result.
← Didn't Know|Knew It →
What is $3\begin{bmatrix}2&-1\0&4\end{bmatrix}$?
What is $3\begin{bmatrix}2&-1\0&4\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}6&-3\0&12\end{bmatrix}$. Multiply each entry by $3$: $3(2)=6$, $3(-1)=-3$, $3(0)=0$, $3(4)=12$.
$\begin{bmatrix}6&-3\0&12\end{bmatrix}$. Multiply each entry by $3$: $3(2)=6$, $3(-1)=-3$, $3(0)=0$, $3(4)=12$.
← Didn't Know|Knew It →
What is the definition of scalar multiplication for a matrix $A$ by a scalar $k$?
What is the definition of scalar multiplication for a matrix $A$ by a scalar $k$?
Tap to reveal answer
$kA$ is the matrix with entries $(kA){ij}=k\cdot a{ij}$. Multiply each entry $a_{ij}$ by the scalar $k$.
$kA$ is the matrix with entries $(kA){ij}=k\cdot a{ij}$. Multiply each entry $a_{ij}$ by the scalar $k$.
← Didn't Know|Knew It →
What happens to each entry of a matrix $A$ when you compute $kA$?
What happens to each entry of a matrix $A$ when you compute $kA$?
Tap to reveal answer
Each entry is multiplied by $k$. Scalar multiplication distributes to every matrix entry.
Each entry is multiplied by $k$. Scalar multiplication distributes to every matrix entry.
← Didn't Know|Knew It →
What is the size of $kA$ if $A$ is an $m\times n$ matrix?
What is the size of $kA$ if $A$ is an $m\times n$ matrix?
Tap to reveal answer
$kA$ is also an $m\times n$ matrix. Scalar multiplication preserves matrix dimensions.
$kA$ is also an $m\times n$ matrix. Scalar multiplication preserves matrix dimensions.
← Didn't Know|Knew It →
State the property: how are $k(A-B)$ and $kA-kB$ related (same-sized $A,B$)?
State the property: how are $k(A-B)$ and $kA-kB$ related (same-sized $A,B$)?
Tap to reveal answer
$k(A-B)=kA-kB$. Scalar multiplication distributes over matrix subtraction.
$k(A-B)=kA-kB$. Scalar multiplication distributes over matrix subtraction.
← Didn't Know|Knew It →
What is $\frac{1}{2}\begin{bmatrix}6&-8\10&0\end{bmatrix}$?
What is $\frac{1}{2}\begin{bmatrix}6&-8\10&0\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}3&-4\5&0\end{bmatrix}$. Multiply each entry by $\frac{1}{2}$: $\frac{1}{2}(6)=3$, etc.
$\begin{bmatrix}3&-4\5&0\end{bmatrix}$. Multiply each entry by $\frac{1}{2}$: $\frac{1}{2}(6)=3$, etc.
← Didn't Know|Knew It →
What is $2\begin{bmatrix}0&-2&5\end{bmatrix}$?
What is $2\begin{bmatrix}0&-2&5\end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix}0&-4&10\end{bmatrix}$. Multiply each entry by $2$: $2(0)=0$, $2(-2)=-4$, $2(5)=10$.
$\begin{bmatrix}0&-4&10\end{bmatrix}$. Multiply each entry by $2$: $2(0)=0$, $2(-2)=-4$, $2(5)=10$.
← Didn't Know|Knew It →