Inverse and Determinant of a Matrix - AP Precalculus

1. What is the determinant of $\begin{bmatrix} 5 & 1 \\ 2 & 3 \end{bmatrix}$?

   13. $(5)(3) - (1)(2) = 15 - 2 = 13$.

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

   $a^2 - b^2$. Cross multiplication gives $a^2 - b^2$.

3. What is the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$?

   -1. $(0)(0) - (1)(1) = 0 - 1 = -1$.

4. What is the determinant of a $3 \times 3$ matrix $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$?

   $a(ei - fh) - b(di - fg) + c(dh - eg)$. Expand along first row using cofactor expansion method.

5. Describe what makes a matrix singular.

   Its determinant is zero. Singular matrices have zero determinant and no inverse.

6. What is the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

   1. This is the $2 \times 2$ identity matrix.

7. What is the determinant of $\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}$?

   21. Product of diagonal elements: $7 \times 3 = 21$.

8. State the determinant of $2 \times 2$ identity matrix $I_2$.

   1. Identity matrices always have unit determinant.

9. What is the result of multiplying a matrix by its inverse?

   The identity matrix. Matrix multiplication with its inverse yields the identity matrix.

10. What is the determinant of a singular matrix?

    0. Singular matrices are defined by having zero determinant.

11. State the rule for the determinant of a triangular matrix.

    It is the product of the diagonal elements. Upper or lower triangular matrices have determinant equal to diagonal product.

12. How does the determinant of $A^T$ compare to $A$?

    They are equal. Transpose operation preserves determinant value.

13. Find the determinant of $\begin{bmatrix} 4 & 5 \\ 7 & 8 \end{bmatrix}$.

    -3. $(4)(8) - (5)(7) = 32 - 35 = -3$.

14. Calculate the determinant of $\begin{bmatrix} 6 & 5 \\ 1 & 3 \end{bmatrix}$.

    13. $(6)(3) - (5)(1) = 18 - 5 = 13$.

15. State the formula to find the inverse of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

    $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Swap $a$ and $d$, negate $b$ and $c$, then divide by determinant.

16. What is the effect on the determinant when a row is multiplied by a scalar $k$?

    It is multiplied by $k$. Scaling a row scales the entire determinant by that factor.

17. Calculate the determinant of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.

    105. Product of diagonal elements: $3 \times 5 \times 7 = 105$.

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

    -2. Using formula: $(1)(4) - (2)(3) = 4 - 6 = -2$.

19. Find the inverse of $\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$.

    $\begin{bmatrix} \frac{1}{3} & 0 \\ 0 & \frac{1}{4} \end{bmatrix}$. Diagonal matrix inverse has reciprocals on the diagonal.

20. State the determinant of $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.

    3. $(2)(2) - (1)(1) = 4 - 1 = 3$.

21. Identify the condition under which a $2 \times 2$ matrix has no inverse.

    When $ad - bc = 0$. Zero determinant means the matrix is singular and non-invertible.

22. What is the adjugate of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$?

    $\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$. Swap diagonal elements and negate off-diagonal elements.

23. If a matrix $A$ is invertible, what can be said about $\det(A)$?

    It is nonzero. Invertible matrices must have nonzero determinant.

24. Find the inverse of $\begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$.

    $\frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix}$. Determinant is $24-14=10$, then apply inverse formula.

25. What is the determinant of $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$?

    $0$. Rows are identical, making matrix singular with zero determinant.

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

    10. $(4)(3) - (2)(1) = 12 - 2 = 10$.

27. Calculate the determinant of $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.

    5. $(2)(4) - (3)(1) = 8 - 3 = 5$.

28. Find the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$.

    2. Product of diagonal elements: $1 \times 2 = 2$.

29. Find the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.

    -1. This is a permutation matrix with determinant $-1$.

30. What is the determinant of an identity matrix?

    1. Identity matrices always have determinant equal to 1.

31. What is the inverse of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?

    $\begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{bmatrix}$. For diagonal matrices, invert each diagonal element.

32. What is the determinant of a zero matrix?

    0. Zero matrices have all zero entries, so determinant is zero.

33. What happens to the determinant if two rows of a matrix are swapped?

    The determinant changes sign. Row swapping introduces a factor of $-1$ to the determinant.

34. What is the result of $\det(AB)$ if $A$ and $B$ are $2 \times 2$ matrices?

    $\det(A) \cdot \det(B)$. Determinant of products equals product of determinants.

35. What is the relationship between the determinants of a matrix and its transpose?

    They are equal. Transposition preserves the determinant value.

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

    $\begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$. Determinant is $-2$, so multiply adjugate by $-\frac{1}{2}$.

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