Scalar Multiplication of Matrices - Precalculus

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

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

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

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

3. What is $1A$ for any matrix $A$?

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

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

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

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

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

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

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

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

   $\begin{bmatrix}8&8\\0&8\end{bmatrix}$. First add matrices: $A+B=\begin{bmatrix}4&4\\0&4\end{bmatrix}$, then multiply by $2$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16. What is $0A$ for any matrix $A$?

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

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

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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