AP Precalculus Flashcards: Inverse And Determinant Of A Matrix

What this deck covers

This deck focuses on Inverse And Determinant Of A Matrix, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

Answer:

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

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

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

Flashcard 3: What is the determinant of a singular matrix?

Answer:

1. Singular matrices are defined by having zero determinant.

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

Answer:

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

Flashcard 5: What is the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$?

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

Flashcard 6: What is the determinant of $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$?

Answer:

1. Rows are identical, making matrix singular with zero determinant.

Flashcard 7: What is the determinant of a $3 \times 3$ matrix $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$?

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

Flashcard 8: Describe what makes a matrix singular.

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

Flashcard 9: What is the effect on the determinant when a row is multiplied by a scalar $k$?

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

Flashcard 10: What is the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

Answer:

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

Flashcard 11: What is the determinant of $\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}$?

Answer:

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

Flashcard 12: State the determinant of $2 \times 2$ identity matrix $I_2$.

Answer:

1. Identity matrices always have unit determinant.

Flashcard 13: What is the result of multiplying a matrix by its inverse?

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

Flashcard 14: What is the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$?

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

Flashcard 15: What is the determinant of a zero matrix?

Answer:

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

Flashcard 16: Find the inverse of $\begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$.

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

Flashcard 17: What is the determinant of a singular matrix?

Answer:

1. Singular matrices are defined by having zero determinant.

Flashcard 18: State the rule for the determinant of a triangular matrix.

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

Flashcard 19: What is the relationship between the determinants of a matrix and its transpose?

Answer: They are equal. Transposition preserves the determinant value.

Flashcard 20: How does the determinant of $A^T$ compare to $A$?

Answer: They are equal. Transpose operation preserves determinant value.

Flashcard 21: What is the determinant of $\begin{bmatrix} a & b \\ b & a \end{bmatrix}$?

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

Flashcard 22: What is the determinant of an identity matrix?

Answer:

1. Identity matrices always have determinant equal to 1.

Flashcard 23: What is the inverse of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?

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

Flashcard 24: Find the determinant of $\begin{bmatrix} 4 & 5 \\ 7 & 8 \end{bmatrix}$.

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

Flashcard 25: Find the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.

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

Flashcard 26: Calculate the determinant of $\begin{bmatrix} 6 & 5 \\ 1 & 3 \end{bmatrix}$.

Answer:

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

Flashcard 27: State the formula to find the inverse of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

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

Flashcard 28: What is the effect on the determinant when a row is multiplied by a scalar $k$?

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

Flashcard 29: Calculate the determinant of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.

Answer:

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

Flashcard 30: Find the determinant of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

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

Flashcard 31: Find the inverse of $\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$.

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

Flashcard 32: State the determinant of $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.

Answer:

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

Flashcard 33: Identify the condition under which a $2 \times 2$ matrix has no inverse.

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

Flashcard 34: What is the adjugate of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$?

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

Flashcard 35: Identify the condition under which a $2 \times 2$ matrix has no inverse.

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

Flashcard 36: If a matrix $A$ is invertible, what can be said about $\det(A)$?

Answer: It is nonzero. Invertible matrices must have nonzero determinant.

Flashcard 37: Calculate the determinant of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.

Answer:

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

Flashcard 38: Find the inverse of $\begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$.

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

Flashcard 39: What is the determinant of $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$?

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

Flashcard 40: What is the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

Answer:

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

Flashcard 41: What happens to the determinant if two rows of a matrix are swapped?

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

Flashcard 42: Find the determinant of $\begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}$.

Answer:

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

Flashcard 43: Calculate the determinant of $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.

Answer:

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

Flashcard 44: State the formula to find the inverse of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

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

Flashcard 45: What is the determinant of a $3 \times 3$ matrix $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$?

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

Flashcard 46: Calculate the determinant of $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.

Answer:

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

Flashcard 47: Find the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$.

Answer:

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

Flashcard 48: What is the result of multiplying a matrix by its inverse?

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

Flashcard 49: Find the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.

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

Flashcard 50: Find the determinant of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

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

Flashcard 51: What is the determinant of an identity matrix?

Answer:

1. Identity matrices always have determinant equal to 1.

Flashcard 52: What is the inverse of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?

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

Flashcard 53: What is the determinant of a zero matrix?

Answer:

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

Flashcard 54: What happens to the determinant if two rows of a matrix are swapped?

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

Flashcard 55: What is the result of $\det(AB)$ if $A$ and $B$ are $2 \times 2$ matrices?

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

Flashcard 56: What is the relationship between the determinants of a matrix and its transpose?

Answer: They are equal. Transposition preserves the determinant value.

Flashcard 57: Find the inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

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