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: 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 2: 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.

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

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

Flashcard 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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.

Flashcard 26: 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 27: 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 28: What is $\det\!\left(\begin{pmatrix}3&1\\2&4\end{pmatrix}\right)$?

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

Flashcard 29: 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 30: 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.