Algebra 2 Flashcards: Solving Systems Using Matrix Inverses

What this deck covers

This deck focuses on Solving Systems Using Matrix Inverses, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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 identity matrix $I_3$?

Answer: $I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$. The 3x3 identity matrix has 1s on the main diagonal and 0s elsewhere, acting as the multiplicative identity.

Flashcard 2: What is $A^{-1}$ for $A=\begin{pmatrix}2&0\\0&5\end{pmatrix}$?

Answer: $A^{-1}=\begin{pmatrix}\frac{1}{2}&0\\0&\frac{1}{5}\end{pmatrix}$. The inverse of a diagonal matrix reciprocates each diagonal entry.

Flashcard 3: What is the relationship between $\det(A)$ and the number of solutions to $A\vec{x}=\vec{b}$?

Answer: $\det(A)\neq 0$ implies a unique solution for every $\vec{b}$. A non-zero determinant confirms invertibility, guaranteeing a unique solution for every $\vec{b}$.

Flashcard 4: Identify whether $A=\begin{pmatrix}1&2\\2&4\end{pmatrix}$ is invertible.

Answer: Not invertible because $\det(A)=0$. A zero determinant confirms the matrix is singular.

Flashcard 5: What does it mean for the columns of $A$ if $A$ is invertible?

Answer: The columns are linearly independent. Invertibility requires the columns to form a linearly independent set.

Flashcard 6: What does it indicate if row reduction of $[A\mid I]$ cannot make the left side $I$?

Answer: $A$ is not invertible, so $A^{-1}$ does not exist. Inability to reach the identity on the left indicates the matrix is singular.

Flashcard 7: What is the augmented matrix form of the system $A\vec{x}=\vec{b}$?

Answer: $[A\mid \vec{b}]$. The augmented matrix combines coefficients and constants for row reduction.

Flashcard 8: What is the identity matrix $I_2$?

Answer: $I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$. The 2x2 identity matrix has 1s on the main diagonal and 0s elsewhere, serving as the multiplicative identity.

Flashcard 9: What matrix equation represents a linear system in matrix form?

Answer: $A\vec{x}=\vec{b}$. This form represents the linear system as a matrix-vector product equaling the constant vector.

Flashcard 10: What is the inverse-matrix solution to $A\vec{x}=\vec{b}$ when $A$ is invertible?

Answer: $\vec{x}=A^{-1}\vec{b}$. Multiplying both sides on the left by $A^{-1}$ isolates $\vec{x}$ as the unique solution.

Flashcard 11: Identify the correct criterion: $A$ is invertible iff $\det(A)=0$ or $\det(A)\neq 0$?

Answer: $A$ is invertible iff $\det(A)\neq 0$. Invertibility requires a non-zero determinant.

Flashcard 12: What is the inverse of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$ when all $d_i\neq 0$?

Answer: $D^{-1}=\mathrm{diag}(\frac{1}{d_1},\dots,\frac{1}{d_n})$. Each diagonal entry is reciprocated to ensure the product is the identity matrix.

Flashcard 13: Find $\det\!\left(\begin{pmatrix}1&0&0\\2&3&0\\4&5&6\end{pmatrix}\right)$.

Answer: $18$. For upper triangular matrices, the determinant is the product of diagonal entries.

Flashcard 14: Identify whether $A=\begin{pmatrix}4&-2\\6&-3\end{pmatrix}$ is invertible.

Answer: Not invertible because $\det(A)=0$. A zero determinant indicates linear dependence and non-invertibility.

Flashcard 15: What does it mean if $A\vec{x}=\vec{b}$ has a unique solution for every $\vec{b}$?

Answer: $A$ is invertible. Invertibility ensures the system has a unique solution for any right-hand side vector.

Flashcard 16: What is the inverse of a scalar multiple $kA$ when $k\neq 0$ and $A$ is invertible?

Answer: $(kA)^{-1}=\frac{1}{k}A^{-1}$. The inverse scales by the reciprocal of $k$ to counteract the scalar multiple.

Flashcard 17: What is the determinant of $A=\begin{pmatrix}a&0\\0&d\end{pmatrix}$?

Answer: $\det(A)=ad$. For diagonal matrices, the determinant is the product of diagonal entries.

Flashcard 18: Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?

Answer: $\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.

Flashcard 19: Compute $(3A)^{-1}$ if $A^{-1}=M$ and $3\neq 0$.

Answer: $(3A)^{-1}=\frac{1}{3}M$. The scalar 3 requires dividing the inverse by 3.

Flashcard 20: What is the determinant test for invertibility of a square matrix $A$?

Answer: $A$ is invertible iff $\det(A)\neq 0$. A non-zero determinant indicates the matrix has full rank and is thus invertible.

Flashcard 21: What is $A^{-1}$ for $A=\begin{pmatrix}1&0&0\\0&-2&0\\0&0&4\end{pmatrix}$?

Answer: $A^{-1}=\begin{pmatrix}1&0&0\\0&-\frac{1}{2}&0\\0&0&\frac{1}{4}\end{pmatrix}$. Reciprocate each diagonal entry to form the inverse.

Flashcard 22: Compute $\det\!\left(\begin{pmatrix}5&0&0\\0&1&0\\0&0&-2\end{pmatrix}\right)$.

Answer: $-10$. The determinant of a diagonal matrix is the product of its diagonal entries.

Flashcard 23: Solve $A\vec{x}=\vec{b}$ if $A^{-1}=\begin{pmatrix}1&0\\-2&1\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}3\\5\end{pmatrix}$.

Answer: $\vec{x}=\begin{pmatrix}3\\-1\end{pmatrix}$. Matrix multiplication of $A^{-1}$ by $\vec{b}$ gives the solution vector.

Flashcard 24: What is $\det\!\left(\begin{pmatrix}3&1\\2&4\end{pmatrix}\right)$?

Answer: $10$. Computed using the 2x2 determinant formula $ad-bc$.

Flashcard 25: What is the inverse of the inverse of an invertible matrix $A$?

Answer: $(A^{-1})^{-1}=A$. Inverting the inverse restores the original matrix, as they are multiplicative inverses.

Flashcard 26: What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?

Answer: The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.

Flashcard 27: Solve using an inverse: $\begin{pmatrix}1&0\\0&2\end{pmatrix}\vec{x}=\begin{pmatrix}5\\6\end{pmatrix}$.

Answer: $\vec{x}=\begin{pmatrix}5\\3\end{pmatrix}$. Multiplying the inverse matrix by the vector yields the solution.

Flashcard 28: Identify the correct statement: $A^{-1}\vec{b}$ or $\vec{b}A^{-1}$ solves $A\vec{x}=\vec{b}$?

Answer: $\vec{x}=A^{-1}\vec{b}$. $A^{-1}\vec{b}$ is the correct order for matrix-vector multiplication to solve the system.

Flashcard 29: What is the inverse of the product $AB$ when $A$ and $B$ are invertible?

Answer: $(AB)^{-1}=B^{-1}A^{-1}$. The inverse of a product requires reversing the order of the individual inverses.

Flashcard 30: What is the determinant of a triangular matrix in terms of its diagonal entries?

Answer: $\det(A)$ equals the product of the diagonal entries. For triangular matrices, the determinant is the product of the diagonal elements.

Flashcard 31: What is the determinant of $A=\begin{pmatrix}a&b\\0&d\end{pmatrix}$?

Answer: $\det(A)=ad$. For upper triangular matrices, the determinant equals the product of diagonals.

Flashcard 32: Which matrix must have the same size as $A$ to form $[A\mid I]$ for an $n\times n$ matrix?

Answer: The identity matrix $I_n$. $I_n$ is square $n\times n$ to match $A$'s dimensions for augmentation.

Flashcard 33: What is $\det(A^{-1})$ in terms of $\det(A)$ for invertible $A$?

Answer: $\det(A^{-1})=\frac{1}{\det(A)}$. This follows from the multiplicative property of determinants and $\det(I)=1$.

Flashcard 34: What is the determinant of the identity matrix $I_n$?

Answer: $\det(I_n)=1$. It is the product of the diagonal entries, all of which are 1.

Flashcard 35: Identify whether $A=\begin{pmatrix}1&0&0\\0&0&2\\0&3&0\end{pmatrix}$ is invertible using $\det(A)$.

Answer: Invertible because $\det(A)=-6\neq 0$. The non-zero determinant confirms the matrix is invertible.

Flashcard 36: Compute $(AB)^{-1}$ if $A^{-1}=P$ and $B^{-1}=Q$ for invertible $A,B$.

Answer: $(AB)^{-1}=QP$. Given $A^{-1}=P$ and $B^{-1}=Q$, the product inverse is $B^{-1}A^{-1}$.

Flashcard 37: What is $\det(AB)$ in terms of $\det(A)$ and $\det(B)$ for square matrices?

Answer: $\det(AB)=\det(A)\det(B)$. The determinant function is multiplicative for square matrices.

Flashcard 38: Find $\left(\begin{pmatrix}0&2\\-3&1\end{pmatrix}\right)^{-1}$.

Answer: $\begin{pmatrix}\frac{1}{6}&-\frac{1}{3}\\frac{1}{2}&0\end{pmatrix}$. The inverse is scaled by $1/\det(A)$ using the adjugate matrix.

Flashcard 39: What is the solution of $A\vec{x}=\vec{0}$ when $A$ is invertible?

Answer: $\vec{x}=\vec{0}$. Invertible $A$ implies only the trivial solution for the homogeneous system.

Flashcard 40: What is $A^{-1}$ if $A$ is the identity matrix $I_n$?

Answer: $I_n^{-1}=I_n$. The identity matrix multiplied by itself yields itself, satisfying the inverse definition.

Flashcard 41: What technology-based method is commonly used to find $A^{-1}$ for $3\times 3$ or larger matrices?

Answer: Use a calculator or software to compute $A^{-1}$ or solve $A\vec{x}=\vec{b}$. Technology is recommended for efficient computation of inverses or solutions for larger matrices.

Flashcard 42: Solve using an inverse: $\begin{pmatrix}2&0\\0&-4\end{pmatrix}\vec{x}=\begin{pmatrix}8\\12\end{pmatrix}$.

Answer: $\vec{x}=\begin{pmatrix}4\\-3\end{pmatrix}$. The inverse scales each equation to solve for the variables.

Flashcard 43: What does it mean if a row of zeros appears on the left side when row-reducing $[A\mid I]$?

Answer: $A$ is singular, so $A^{-1}$ does not exist. A zero row indicates linear dependence, meaning the matrix is singular.

Flashcard 44: What does $A\vec{x}=\vec{b}$ represent when $A$ is $3\times 3$?

Answer: A system of $3$ linear equations in $3$ unknowns. The dimensions of $A$ determine the number of equations and unknowns.

Flashcard 45: What does it mean for a square matrix $A$ to be invertible?

Answer: $A^{-1}$ exists such that $AA^{-1}=I$ and $A^{-1}A=I$. A square matrix is invertible if there exists a unique matrix that, when multiplied on either side, yields the identity matrix.

Flashcard 46: What condition on $a,b,c,d$ makes $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ non-invertible?

Answer: $ad-bc=0$. This condition makes the determinant zero, rendering the matrix singular and non-invertible.

Flashcard 47: What is the determinant of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$?

Answer: $\det(D)=d_1d_2\cdots d_n$. The determinant is the product of its diagonal entries.

Flashcard 48: What matrix is used to compute $A^{-1}$ by row reduction for an $n\times n$ matrix $A$?

Answer: Augment $[A\mid I_n]$ and row-reduce to $[I_n\mid A^{-1}]$. Row operations transform $A$ to $I_n$, applying the same to $I_n$ to obtain $A^{-1}$.

Flashcard 49: Identify the matrix equation you should solve to find $A^{-1}$ using row reduction.

Answer: Row-reduce $[A\mid I_n]$ to $[I_n\mid A^{-1}]$. Row reduction of the augmented matrix yields the inverse on the right.

Flashcard 50: What is the inverse of a $2\times 2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ when it exists?

Answer: $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. The formula scales the adjugate matrix by the reciprocal of the determinant to satisfy $AA^{-1}=I$.

Flashcard 51: What is the transpose-inverse rule for an invertible matrix $A$?

Answer: $(A^T)^{-1}=(A^{-1})^T$. The transpose of the inverse equals the inverse of the transpose for invertible matrices.

Flashcard 52: What equation defines the inverse matrix $A^{-1}$ of a square matrix $A$?

Answer: $AA^{-1}=I$ (equivalently $A^{-1}A=I$). The inverse satisfies the multiplicative identity property when multiplied on the left or right by the original matrix.

Flashcard 53: Compute $\det\!\left(\begin{pmatrix}0&2\\-3&1\end{pmatrix}\right)$.

Answer: $6$. Apply the 2x2 determinant formula $ad-bc$.

Flashcard 54: Find $\left(\begin{pmatrix}1&2\\3&4\end{pmatrix}\right)^{-1}$.

Answer: $\begin{pmatrix}-2&1\\frac{3}{2}&-\frac{1}{2}\end{pmatrix}$. The 2x2 inverse formula scales the adjugate by $1/\det(A)$.

Flashcard 55: What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?

Answer: $A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.