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 $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

Answer:

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

Flashcard 2: 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 3: If a matrix $A$ is invertible, what can be said about $\det(A)$?

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

Flashcard 4: 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 5: 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 6: State the determinant of $2 \times 2$ identity matrix $I_2$.

Answer:

1. Identity matrices always have unit determinant.

Flashcard 7: How does the determinant of $A^T$ compare to $A$?

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

Flashcard 8: What is the determinant of a zero matrix?

Answer:

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

Flashcard 9: 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 10: Describe what makes a matrix singular.

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

Flashcard 11: What is the relationship between the determinants of a matrix and its transpose?

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

Flashcard 12: 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 13: What is the determinant of an identity matrix?

Answer:

1. Identity matrices always have determinant equal to 1.

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: What is the determinant of a singular matrix?

Answer:

1. Singular matrices are defined by having zero determinant.

Flashcard 20: 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 21: What is the determinant of a singular matrix?

Answer:

1. Singular matrices are defined by having zero determinant.

Flashcard 22: What is the relationship between the determinants of a matrix and its transpose?

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

Flashcard 23: 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 24: What is the determinant of a zero matrix?

Answer:

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

Flashcard 25: What is the determinant of an identity matrix?

Answer:

1. Identity matrices always have determinant equal to 1.

Flashcard 26: 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 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 determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

Answer:

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

Flashcard 29: 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 30: What is the determinant of $\begin{bmatrix} a & b \\ b & a \end{bmatrix}$?

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