Matrix Addition, Subtraction, and Multiplication - Precalculus

1. What is $A+B$ for $A=\begin{pmatrix}1&-2\\3&0\end{pmatrix}$ and $B=\begin{pmatrix}4&5\\-1&2\end{pmatrix}$?

   $\begin{pmatrix}5&3\\2&2\end{pmatrix}$. Add corresponding entries: $(1+4, -2+5; 3+(-1), 0+2)$.

2. What is $3A$ if $A=\begin{bmatrix}-1&2\\0&5\end{bmatrix}$?

   $\begin{bmatrix}-3&6\\0&15\end{bmatrix}$. Multiply each entry by 3: $3(-1)=-3$, $3(2)=6$, $3(0)=0$, $3(5)=15$.

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

   $\begin{bmatrix}5&3\\2&2\end{bmatrix}$. Add corresponding entries: $1+4=5$, $-2+5=3$, $3+(-1)=2$, $0+2=2$.

4. Which statement is true about commutativity of matrix multiplication: $AB=BA$ always?

   False; in general $AB\ne BA$ (and one product may be undefined). Matrix multiplication is not commutative.

5. What condition on dimensions must be true to subtract matrices $A-B$?

   $A$ and $B$ must have the same dimensions $m \times n$. Subtraction requires matching rows and columns to subtract corresponding entries.

6. What is the required dimension condition for the product $AB$ to be defined?

   If $A$ is $m\times n$ and $B$ is $n\times p$. Number of columns in $A$ must equal number of rows in $B$.

7. What is the multiplicative identity for an $n\times n$ matrix $A$?

   The identity matrix $I_n$, so $AI_n=I_nA=A$. Identity matrix leaves any square matrix unchanged.

8. What is $(AB)C$ compared to $A(BC)$ when all products are defined?

   $(AB)C=A(BC)$. Matrix multiplication is associative.

9. What is the entrywise rule for matrix subtraction $(A-B)_{ij}$?

   $(A-B)_{ij}=a_{ij}-b_{ij}$. Subtract corresponding entries at the same position in both matrices.

10. What is the identity matrix property for multiplication with $I_n$?

    $I_nA=A$ and $AI_n=A$ (when dimensions match). Identity matrix acts as multiplicative identity element.

11. What condition on dimensions is required to add matrices $A$ and $B$?

    $A$ and $B$ must have the same dimensions ($m\times n$). Addition requires corresponding entries to exist.

12. What is $AB$ if $A=\begin{bmatrix}1\\-2\\3\end{bmatrix}$ and $B=\begin{bmatrix}4&0\end{bmatrix}$?

    $\begin{bmatrix}4&0\\-8&0\\12&0\end{bmatrix}$. Column times row gives outer product matrix.

13. What is the dot-product formula for an entry $(AB)_{ij}$?

    $(AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot product with column $j$ of $B$.

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

    $2$. Row vector times column vector gives scalar: $2(3)+(-1)(4)+0(5)=2$.

15. What is the scalar multiplication rule for entries of $cA$?

    $(cA)_{ij}=c\,a_{ij}$. Multiply each entry by the scalar $c$.

16. What is the entrywise rule for matrix addition $(A+B)_{ij}$?

    $(A+B)_{ij}=a_{ij}+b_{ij}$. Add corresponding entries at the same position in both matrices.

17. What is $A-B$ if $A=\begin{bmatrix}6&1\\-2&4\end{bmatrix}$ and $B=\begin{bmatrix}3&7\\5&-1\end{bmatrix}$?

    $\begin{bmatrix}3&-6\\-7&5\end{bmatrix}$. Subtract corresponding entries: $6-3=3$, $1-7=-6$, $-2-5=-7$, $4-(-1)=5$.

18. What is $AB$ for $A=\begin{pmatrix}1&0\\-2&3\end{pmatrix}$ and $B=\begin{pmatrix}5\\-1\end{pmatrix}$?

    $\begin{pmatrix}5\\-13\end{pmatrix}$. Row 1: $(1)(5)+(0)(-1)=5$; Row 2: $(-2)(5)+(3)(-1)=-10-3=-13$.

19. What is $AB$ for $A=\begin{pmatrix}2&-1&0\end{pmatrix}$ and $B=\begin{pmatrix}3\\4\\-2\end{pmatrix}$?

    $2$. $(2)(3)+(-1)(4)+(0)(-2)=6-4+0=2$.

20. What is the additive identity matrix for addition with an $m\times n$ matrix $A$?

    The $m\times n$ zero matrix $0$, so $A+0=A$. Adding zero matrix leaves any matrix unchanged.

21. What condition on dimensions must be true to multiply matrices $AB$?

    If $A$ is $m\times n$, then $B$ must be $n\times p$. Number of columns in $A$ must equal number of rows in $B$.

22. What is $A+(-A)$ for any matrix $A$?

    The zero matrix of the same size as $A$. A matrix plus its negative gives all zero entries.

23. What is the distributive property of matrix multiplication over addition?

    $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$. Multiplication distributes over addition from both sides.

24. If $A$ is $m\times n$ and $B$ is $n\times p$, what is the size of $AB$?

    $AB$ is $m\times p$. Product has rows of $A$ and columns of $B$.

25. What is $AB$ for $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}0&1\\-1&2\end{pmatrix}$?

    $\begin{pmatrix}-2&5\\-4&11\end{pmatrix}$. Row 1: $(1)(0)+(2)(-1)=-2$, $(1)(1)+(2)(2)=5$; Row 2: $(3)(0)+(4)(-1)=-4$, $(3)(1)+(4)(2)=11$.

26. What is $AB$ if $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}0&1\\-1&2\end{bmatrix}$?

    $\begin{bmatrix}-2&5\\-4&11\end{bmatrix}$. Row-column products: $(1,2)\cdot(0,-1)=-2$, $(1,2)\cdot(1,2)=5$, etc.

27. What is $I_2\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

    $\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Identity matrix preserves all entries when multiplied.

28. What condition on dimensions must be true to add matrices $A$ and $B$?

    $A$ and $B$ must have the same dimensions $m \times n$. Addition requires matching rows and columns to add corresponding entries.

29. What is the size (dimensions) of $AB$ if $A$ is $m\times n$ and $B$ is $n\times p$?

    $AB$ is $m\times p$. Product has $A$'s rows and $B$'s columns.

30. What distributive property relates multiplication over addition: $A(B+C)$?

    $A(B+C)=AB+AC$ (when dimensions are compatible). Matrix multiplication distributes over addition.

31. What is the formula for an entry of a product matrix $(AB)_{ij}$?

    $(AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ of $A$ with column $j$ of $B$.

32. What is the commutative property status for matrix multiplication?

    In general, $AB\ne BA$ even when both products exist. Matrix multiplication is not commutative unlike scalar multiplication.

33. What is $A-B$ for $A=\begin{pmatrix}7&1\\-3&4\end{pmatrix}$ and $B=\begin{pmatrix}2&-5\\6&0\end{pmatrix}$?

    $\begin{pmatrix}5&6\\-9&4\end{pmatrix}$. Subtract corresponding entries: $(7-2, 1-(-5); -3-6, 4-0)$.

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