Precalculus Flashcards: Scalar Multiplication Of Matrices

What this deck covers

This deck focuses on Scalar Multiplication Of Matrices, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the distributive property: how are $k(A+B)$ and $kA+kB$ related?

Answer: $k(A+B)=kA+kB$. Scalar multiplication distributes over matrix addition.

Flashcard 2: What is $-\frac{3}{2}\begin{bmatrix}2&-4\\-6&8\end{bmatrix}$?

Answer: $\begin{bmatrix}-3&6\\9&-12\end{bmatrix}$. Multiply each entry by $-\frac{3}{2}$.

Flashcard 3: What is $1A$ for any matrix $A$?

Answer: $A$. The multiplicative identity scalar leaves the matrix unchanged.

Flashcard 4: State the property: how are $(k+m)A$ and $kA+mA$ related?

Answer: $(k+m)A=kA+mA$. Scalar addition distributes over matrix multiplication.

Flashcard 5: State the property: how are $k(mA)$ and $(km)A$ related?

Answer: $k(mA)=(km)A$. Scalar multiplication is associative.

Flashcard 6: State the property: how are $k(A-B)$ and $kA-kB$ related (same-sized $A,B$)?

Answer: $k(A-B)=kA-kB$. Scalar multiplication distributes over matrix subtraction.

Flashcard 7: What is $2(A+B)$ if $A=\begin{bmatrix}1&0\\2&-1\end{bmatrix}$ and $B=\begin{bmatrix}3&4\\-2&5\end{bmatrix}$?

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 $2$.

Flashcard 8: What is $-2\begin{bmatrix}5\\-3\\1\end{bmatrix}$?

Answer: $\begin{bmatrix}-10\\6\\-2\end{bmatrix}$. Multiply each entry by $-2$: $-2(5)=-10$, $-2(-3)=6$, $-2(1)=-2$.

Flashcard 9: Find the missing entry $x$ if $-3\begin{bmatrix}2&x\end{bmatrix}=\begin{bmatrix}-6&9\end{bmatrix}$.

Answer: $x=-3$. Since $-3(x)=9$, divide: $x=9/(-3)=-3$.

Flashcard 10: Compute the doubled payoff matrix of $P=\begin{bmatrix}-1&4\\0&2\end{bmatrix}$.

Answer: $\begin{bmatrix}-2&8\\0&4\end{bmatrix}$. Multiply each payoff by 2: $2(-1)=-2$, $2(4)=8$, $2(0)=0$, $2(2)=4$.

Flashcard 11: What is $(k+m)A$ if $A=\begin{bmatrix}1&-2\\0&3\end{bmatrix}$, $k=2$, and $m=-5$?

Answer: $\begin{bmatrix}-3&6\\0&-9\end{bmatrix}$. Since $k+m=2+(-5)=-3$, multiply each entry by $-3$.

Flashcard 12: What is $2\begin{bmatrix}0&-2&5\end{bmatrix}$?

Answer: $\begin{bmatrix}0&-4&10\end{bmatrix}$. Multiply each entry by $2$: $2(0)=0$, $2(-2)=-4$, $2(5)=10$.

Flashcard 13: What is the definition of scalar multiplication for a matrix $A$ by a scalar $k$?

Answer: $kA$ is the matrix with entries $(kA)_{ij}=k\cdot a_{ij}$. Multiply each entry $a_{ij}$ by the scalar $k$.

Flashcard 14: What is $\frac{1}{2}\begin{bmatrix}8&-6\end{bmatrix}$?

Answer: $\begin{bmatrix}4&-3\end{bmatrix}$. Multiply by $\frac{1}{2}$: $\frac{1}{2}(8)=4$, $\frac{1}{2}(-6)=-3$.

Flashcard 15: What is $\frac{1}{2}\begin{bmatrix}6&-8\\10&0\end{bmatrix}$?

Answer: $\begin{bmatrix}3&-4\\5&0\end{bmatrix}$. Multiply each entry by $\frac{1}{2}$: $\frac{1}{2}(6)=3$, etc.

Flashcard 16: What is $0A$ for any matrix $A$?

Answer: The zero matrix of the same size as $A$. Multiplying by zero scalar gives all zero entries.

Flashcard 17: Find and correct the error: $2\begin{bmatrix}1&3\\2&4\end{bmatrix}=\begin{bmatrix}2&3\\2&8\end{bmatrix}$.

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.

Flashcard 18: State the rule for scalar multiplication of a matrix $A$ by a scalar $k$.

Answer: $kA=[k a_{ij}]$ (multiply every entry $a_{ij}$ by $k$). Each element is multiplied by the scalar.

Flashcard 19: What is $-2\begin{bmatrix}1&5&0\\-3&2&7\end{bmatrix}$?

Answer: $\begin{bmatrix}-2&-10&0\\6&-4&-14\end{bmatrix}$. Multiply each entry by $-2$ to get the result.

Flashcard 20: What is $3\begin{bmatrix}1&-2\\0&4\end{bmatrix}$?

Answer: $\begin{bmatrix}3&-6\\0&12\end{bmatrix}$. Multiply each entry: $3(1)=3$, $3(-2)=-6$, $3(0)=0$, $3(4)=12$.

Flashcard 21: State the distributive property: how are $(k+m)A$ and $kA+mA$ related?

Answer: $(k+m)A=kA+mA$. Scalar addition distributes over scalar multiplication.

Flashcard 22: What is $4\begin{bmatrix}-1\\3\\0\end{bmatrix}$?

Answer: $\begin{bmatrix}-4\\12\\0\end{bmatrix}$. Multiply each entry by $4$: $4(-1)=-4$, $4(3)=12$, $4(0)=0$.

Flashcard 23: Compute $(3+(-1))A$ for $A=\begin{bmatrix}2&-4\\1&0\end{bmatrix}$.

Answer: $\begin{bmatrix}4&-8\\2&0\end{bmatrix}$. $(3+(-1))=2$, so multiply each entry by 2.

Flashcard 24: State the property: how are $k(A+B)$ and $kA+kB$ related (same-sized $A,B$)?

Answer: $k(A+B)=kA+kB$. Scalar multiplication distributes over matrix addition.

Flashcard 25: Identify the scalar $k$ if $k\begin{bmatrix}2&-1\end{bmatrix}=\begin{bmatrix}10&-5\end{bmatrix}$.

Answer: $k=5$. Since $5(2)=10$ and $5(-1)=-5$, the scalar is $k=5$.

Flashcard 26: What is $(-1)A$ for any matrix $A$?

Answer: $(-1)A=-A$ (negate every entry). Multiplying by $-1$ changes the sign of every entry.

Flashcard 27: Identify the scalar $k$ if $k\begin{bmatrix}2&-1\end{bmatrix}=\begin{bmatrix}6&-3\end{bmatrix}$.

Answer: $k=3$. Since $k(2)=6$ and $k(-1)=-3$, we have $k=3$.

Flashcard 28: Compute $-1\cdot\begin{bmatrix}-2&0\\5&-7\end{bmatrix}$.

Answer: $\begin{bmatrix}2&0\\-5&7\end{bmatrix}$. Change signs: $-1(-2)=2$, $-1(0)=0$, $-1(5)=-5$, $-1(-7)=7$.

Flashcard 29: What happens to every payoff in a payoff matrix $P$ when you double the game's payoffs?

Answer: The new matrix is $2P$. Doubling payoffs means multiplying the payoff matrix by 2.

Flashcard 30: A payoff matrix is $P=\begin{bmatrix}1&-2\\3&0\end{bmatrix}$. What is the doubled payoff matrix $2P$?

Answer: $\begin{bmatrix}2&-4\\6&0\end{bmatrix}$. Double each payoff by multiplying each entry by $2$.

Flashcard 31: What is $3\begin{bmatrix}2&-1\\0&4\end{bmatrix}$?

Answer: $\begin{bmatrix}6&-3\\0&12\end{bmatrix}$. Multiply each entry by $3$: $3(2)=6$, $3(-1)=-3$, $3(0)=0$, $3(4)=12$.

Flashcard 32: What is the size of $kA$ if $A$ is an $m\times n$ matrix?

Answer: $kA$ is also an $m\times n$ matrix. Scalar multiplication preserves matrix dimensions.

Flashcard 33: What is the size of $kA$ if $A$ is an $m\times n$ matrix and $k$ is a scalar?

Answer: $kA$ is still an $m\times n$ matrix. Scalar multiplication doesn't change matrix dimensions.

Flashcard 34: What is $0\cdot\begin{bmatrix}2&7\\-1&3\end{bmatrix}$?

Answer: $\begin{bmatrix}0&0\\0&0\end{bmatrix}$. Zero times any matrix gives the zero matrix.

Flashcard 35: What is $1\cdot A$ for any matrix $A$?

Answer: $1A=A$. Multiplying by 1 leaves the matrix unchanged (identity property).

Flashcard 36: State the associative property for scalars: how are $k(mA)$ and $(km)A$ related?

Answer: $k(mA)=(km)A$. Scalar multiplication is associative.

Flashcard 37: Compute $4\left(\frac{1}{2}A\right)$ for $A=\begin{bmatrix}6&-2\end{bmatrix}$.

Answer: $\begin{bmatrix}12&-4\end{bmatrix}$. First: $\frac{1}{2}A=\begin{bmatrix}3&-1\end{bmatrix}$, then multiply by 4.

Flashcard 38: What happens to each entry of a matrix $A$ when you compute $kA$?

Answer: Each entry is multiplied by $k$. Scalar multiplication distributes to every matrix entry.