Matrices - ACT Math

1. Identify the order of a $3 \times 2$ matrix.

   3 rows and 2 columns. The first number is rows, second is columns.

2. What is an invertible matrix?

   A square matrix with a non-zero determinant. Non-zero determinant means the matrix has an inverse.

3. Define a zero matrix.

   A matrix where all elements are zero. Acts as the additive identity in matrix operations.

4. Find the transpose of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

   $\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. Each row becomes a column in the transposed matrix.

5. What is matrix addition?

   Adding corresponding elements of two matrices of the same order. Both matrices must have identical dimensions.

6. What does it mean if two matrices are equal?

   They have the same dimensions and corresponding elements are equal. Same size and all corresponding entries match exactly.

7. What is an orthogonal matrix?

   A square matrix whose transpose is also its inverse. Special property where $A^T = A^{-1}$.

8. Find the trace of $\begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}$.

   10. Sum the diagonal elements: $3 + 7 = 10$.

9. What is the order of a matrix?

   The dimensions of a matrix, given as rows $\times$ columns. Specifies the size and shape of the matrix.

10. Define the inverse of a matrix.

    A matrix that, when multiplied with the original, results in the identity matrix. Denoted as $A^{-1}$ where $A \cdot A^{-1} = I$.

11. What is a diagonal matrix?

    A square matrix where all off-diagonal elements are zero. Non-zero elements only appear along the main diagonal.

12. What is matrix subtraction?

    Subtracting corresponding elements of two matrices of the same order. Subtract element-wise from corresponding positions.

13. Calculate $2 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

    $\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Each element multiplied by $2$: $(2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4)$.

14. How do you denote a matrix?

    Typically with uppercase letters, e.g., $A$, $B$, $C$. Standard mathematical convention for matrix notation.

15. What is the condition for matrix multiplication to be defined?

    Number of columns in the first matrix must equal the number of rows in the second. Inner dimensions must match for multiplication to work.

16. What is the determinant of $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

    $ad - bc$. Formula for $2 \times 2$ matrix determinant calculation.

17. Calculate the sum of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.

    $\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$. Add element-wise: $(1+5, 2+6, 3+7, 4+8)$.

18. Is $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ symmetric?

    No. Not symmetric because $a_{12} \neq a_{21}$ (2 ≠ 3).

19. What is an identity matrix?

    A square matrix with 1s on the diagonal and 0s elsewhere. Acts as the multiplicative identity in matrix operations.

20. What does the identity matrix do in multiplication?

    Leaves any matrix unchanged when multiplied. Multiplicative identity property: $AI = IA = A$.

21. Find the transpose of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

    $\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. Each row becomes a column in the transposed matrix.

22. What is a singular matrix?

    A square matrix with a determinant of zero. Cannot be inverted due to zero determinant.

23. What is the trace of a matrix?

    The sum of the diagonal elements of a square matrix. Add all diagonal elements from top-left to bottom-right.

24. What is a symmetric matrix?

    A matrix that is equal to its transpose. Mirror symmetry across the main diagonal.

25. Calculate the sum of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.

    $\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$. Add element-wise: $(1+5, 2+6, 3+7, 4+8)$.

26. What is a matrix?

    A rectangular array of numbers arranged in rows and columns. Numbers organized in a grid format for mathematical operations.

27. Which matrix operation is not commutative?

    Matrix multiplication. Order matters in matrix multiplication: $AB \neq BA$.

28. What is a square matrix?

    A matrix with the same number of rows and columns. Equal dimensions create a special matrix type.

29. Calculate $2 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}$

    $\begin{bmatrix} 2 & 4 \\ 6 & 8 \\ \end{bmatrix}$. Each element multiplied by $2$: $(2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4)$

30. What is the determinant of $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

    $ad - bc$. Formula for $2 \times 2$ matrix determinant calculation.

31. What is a column matrix?

    A matrix with a single column. Also called a column vector in linear algebra.

32. What is a row matrix?

    A matrix with a single row. Also called a row vector in linear algebra.

33. What is a transpose of a matrix?

    Switching the rows and columns of a matrix. Rows become columns and columns become rows.

34. What is matrix multiplication?

    Dot product of rows and columns from two matrices. Row elements multiplied by column elements, then summed.

35. What is a determinant?

    A scalar value that is a function of a square matrix. Measures certain properties of square matrices.

36. Find the trace of $\begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}$.

    10. Sum the diagonal elements: $3 + 7 = 10$.

37. What is scalar multiplication in matrices?

    Multiplying each element of a matrix by a scalar. Multiply every element by the same scalar value.

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