Matrices as Functions - AP Precalculus

1. Define a square matrix.

   A matrix with the same number of rows and columns. Equal dimensions make it square-shaped.

2. Identify the transpose of matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

   $A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. Transpose switches rows and columns.

3. What is the commutative property of matrix multiplication?

   Matrix multiplication is not commutative. Generally $AB ≠ BA$ for matrices.

4. State the order of a matrix with 3 rows and 4 columns.

   The order is $3 \times 4$. Order is written as rows × columns.

5. What condition makes a $2 \times 2$ matrix singular?

   The determinant is zero. Zero determinant means no inverse exists.

6. What is matrix addition?

   The sum of two matrices by adding corresponding elements. Add elements in same positions.

7. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A - A$.

   $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Subtracting gives the zero matrix.

8. What condition is necessary for matrix multiplication $AB$?

   The number of columns in $A$ must equal the number of rows in $B$. Inner dimensions must match for multiplication.

9. What is a matrix?

   A rectangular array of numbers arranged in rows and columns. Basic definition of a matrix structure.

10. Find the determinant of $\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}$.

    The determinant is $7$. Calculate: $(3)(5) - (4)(2) = 15 - 8 = 7$.

11. What is a symmetric matrix?

    A matrix where $A = A^T$. Matrix equals its own transpose.

12. What is the associative property of matrix multiplication?

    $(AB)C = A(BC)$. Grouping doesn't affect the product.

13. What is the element in the second row, third column of matrix $A$?

    $a_{2,3}$. Standard notation: $a_{row,column}$.

14. What is matrix multiplication?

    The product of matrices by row-by-column multiplication. Dot product of rows and columns.

15. Is $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ symmetric?

    Yes, it is symmetric. Check if matrix equals its transpose.

16. What is a diagonal matrix?

    A matrix where all off-diagonal elements are zero. Non-diagonal entries are all zero.

17. What is a column matrix?

    A matrix with only one column. Single vertical array of elements.

18. Define scalar multiplication of a matrix.

    Multiplying each element of a matrix by a scalar. Scale each entry by the scalar value.

19. Define zero matrix.

    A matrix in which all elements are zero. All entries equal zero.

20. What is the identity property of matrix multiplication?

    $AI = IA = A$, where $I$ is the identity matrix. Identity matrix preserves multiplication.

21. What is the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

    $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Formula for 2×2 matrix inverse.

22. What condition is necessary for two matrices to be added?

    They must have the same dimensions. Matrices must be same size to add.

23. What is the trace of a matrix?

    The sum of the diagonal elements of a square matrix. Sum of main diagonal elements.

24. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, what is $2A$?

    $2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Multiply each element by the scalar 2.

25. What is the determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

    $ad - bc$. Cross-multiply and subtract for 2×2.

26. What is the result of $A + 0$ for any matrix $A$?

    The matrix $A$ itself. Zero matrix is additive identity.

27. Find the trace of $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$.

    The trace is $15$. Add diagonal elements: $4 + 5 + 6 = 15$.

28. What is an invertible matrix?

    A matrix with a non-zero determinant and an inverse. Has an inverse that undoes multiplication.

29. What is a row matrix?

    A matrix with only one row. Single horizontal array of elements.

30. What is an antisymmetric matrix?

    A matrix where $A = -A^T$. Matrix equals negative of its transpose.

31. What is the identity matrix of order 2?

    $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Diagonal of ones, zeros elsewhere.

32. Is $\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$ antisymmetric?

    Yes, it is antisymmetric. Check if matrix equals negative transpose.

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