Solving Systems Using Matrix Inverses - Algebra 2
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Solve using an inverse: $\begin{pmatrix}1&0\0&2\end{pmatrix}\vec{x}=\begin{pmatrix}5\6\end{pmatrix}$.
Solve using an inverse: $\begin{pmatrix}1&0\0&2\end{pmatrix}\vec{x}=\begin{pmatrix}5\6\end{pmatrix}$.
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$\vec{x}=\begin{pmatrix}5\3\end{pmatrix}$. Multiplying the inverse matrix by the vector yields the solution.
$\vec{x}=\begin{pmatrix}5\3\end{pmatrix}$. Multiplying the inverse matrix by the vector yields the solution.
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What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?
What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?
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$A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.
$A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.
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What is $\det!\left(\begin{pmatrix}3&1\2&4\end{pmatrix}\right)$?
What is $\det!\left(\begin{pmatrix}3&1\2&4\end{pmatrix}\right)$?
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$10$. Computed using the 2x2 determinant formula $ad-bc$.
$10$. Computed using the 2x2 determinant formula $ad-bc$.
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Which matrix must have the same size as $A$ to form $[A\mid I]$ for an $n\times n$ matrix?
Which matrix must have the same size as $A$ to form $[A\mid I]$ for an $n\times n$ matrix?
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The identity matrix $I_n$. $I_n$ is square $n\times n$ to match $A$'s dimensions for augmentation.
The identity matrix $I_n$. $I_n$ is square $n\times n$ to match $A$'s dimensions for augmentation.
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Identify whether $A=\begin{pmatrix}1&0&0\0&0&2\0&3&0\end{pmatrix}$ is invertible using $\det(A)$.
Identify whether $A=\begin{pmatrix}1&0&0\0&0&2\0&3&0\end{pmatrix}$ is invertible using $\det(A)$.
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Invertible because $\det(A)=-6\neq 0$. The non-zero determinant confirms the matrix is invertible.
Invertible because $\det(A)=-6\neq 0$. The non-zero determinant confirms the matrix is invertible.
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Find $\left(\begin{pmatrix}0&2\-3&1\end{pmatrix}\right)^{-1}$.
Find $\left(\begin{pmatrix}0&2\-3&1\end{pmatrix}\right)^{-1}$.
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$\begin{pmatrix}\frac{1}{6}&-\frac{1}{3}\frac{1}{2}&0\end{pmatrix}$. The inverse is scaled by $1/\det(A)$ using the adjugate matrix.
$\begin{pmatrix}\frac{1}{6}&-\frac{1}{3}\frac{1}{2}&0\end{pmatrix}$. The inverse is scaled by $1/\det(A)$ using the adjugate matrix.
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What does $A\vec{x}=\vec{b}$ represent when $A$ is $3\times 3$?
What does $A\vec{x}=\vec{b}$ represent when $A$ is $3\times 3$?
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A system of $3$ linear equations in $3$ unknowns. The dimensions of $A$ determine the number of equations and unknowns.
A system of $3$ linear equations in $3$ unknowns. The dimensions of $A$ determine the number of equations and unknowns.
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Identify the matrix equation you should solve to find $A^{-1}$ using row reduction.
Identify the matrix equation you should solve to find $A^{-1}$ using row reduction.
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Row-reduce $[A\mid I_n]$ to $[I_n\mid A^{-1}]$. Row reduction of the augmented matrix yields the inverse on the right.
Row-reduce $[A\mid I_n]$ to $[I_n\mid A^{-1}]$. Row reduction of the augmented matrix yields the inverse on the right.
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Identify the correct criterion: $A$ is invertible iff $\det(A)=0$ or $\det(A)\neq 0$?
Identify the correct criterion: $A$ is invertible iff $\det(A)=0$ or $\det(A)\neq 0$?
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$A$ is invertible iff $\det(A)\neq 0$. Invertibility requires a non-zero determinant.
$A$ is invertible iff $\det(A)\neq 0$. Invertibility requires a non-zero determinant.
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Compute $(3A)^{-1}$ if $A^{-1}=M$ and $3\neq 0$.
Compute $(3A)^{-1}$ if $A^{-1}=M$ and $3\neq 0$.
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$(3A)^{-1}=\frac{1}{3}M$. The scalar 3 requires dividing the inverse by 3.
$(3A)^{-1}=\frac{1}{3}M$. The scalar 3 requires dividing the inverse by 3.
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What equation defines the inverse matrix $A^{-1}$ of a square matrix $A$?
What equation defines the inverse matrix $A^{-1}$ of a square matrix $A$?
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$AA^{-1}=I$ (equivalently $A^{-1}A=I$). The inverse satisfies the multiplicative identity property when multiplied on the left or right by the original matrix.
$AA^{-1}=I$ (equivalently $A^{-1}A=I$). The inverse satisfies the multiplicative identity property when multiplied on the left or right by the original matrix.
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What is $A^{-1}$ for $A=\begin{pmatrix}1&0&0\0&-2&0\0&0&4\end{pmatrix}$?
What is $A^{-1}$ for $A=\begin{pmatrix}1&0&0\0&-2&0\0&0&4\end{pmatrix}$?
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$A^{-1}=\begin{pmatrix}1&0&0\0&-\frac{1}{2}&0\0&0&\frac{1}{4}\end{pmatrix}$. Reciprocate each diagonal entry to form the inverse.
$A^{-1}=\begin{pmatrix}1&0&0\0&-\frac{1}{2}&0\0&0&\frac{1}{4}\end{pmatrix}$. Reciprocate each diagonal entry to form the inverse.
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What is the inverse of a $2\times 2$ matrix $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ when it exists?
What is the inverse of a $2\times 2$ matrix $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ when it exists?
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$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. The formula scales the adjugate matrix by the reciprocal of the determinant to satisfy $AA^{-1}=I$.
$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. The formula scales the adjugate matrix by the reciprocal of the determinant to satisfy $AA^{-1}=I$.
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What is the determinant of the identity matrix $I_n$?
What is the determinant of the identity matrix $I_n$?
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$\det(I_n)=1$. It is the product of the diagonal entries, all of which are 1.
$\det(I_n)=1$. It is the product of the diagonal entries, all of which are 1.
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Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?
Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?
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$\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.
$\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.
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What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?
What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?
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The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.
The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.
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What is the solution of $A\vec{x}=\vec{0}$ when $A$ is invertible?
What is the solution of $A\vec{x}=\vec{0}$ when $A$ is invertible?
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$\vec{x}=\vec{0}$. Invertible $A$ implies only the trivial solution for the homogeneous system.
$\vec{x}=\vec{0}$. Invertible $A$ implies only the trivial solution for the homogeneous system.
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What is the determinant of $A=\begin{pmatrix}a&b\0&d\end{pmatrix}$?
What is the determinant of $A=\begin{pmatrix}a&b\0&d\end{pmatrix}$?
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$\det(A)=ad$. For upper triangular matrices, the determinant equals the product of diagonals.
$\det(A)=ad$. For upper triangular matrices, the determinant equals the product of diagonals.
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What is the inverse of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$ when all $d_i\neq 0$?
What is the inverse of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$ when all $d_i\neq 0$?
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$D^{-1}=\mathrm{diag}(\frac{1}{d_1},\dots,\frac{1}{d_n})$. Each diagonal entry is reciprocated to ensure the product is the identity matrix.
$D^{-1}=\mathrm{diag}(\frac{1}{d_1},\dots,\frac{1}{d_n})$. Each diagonal entry is reciprocated to ensure the product is the identity matrix.
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What is the determinant of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$?
What is the determinant of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$?
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$\det(D)=d_1d_2\cdots d_n$. The determinant is the product of its diagonal entries.
$\det(D)=d_1d_2\cdots d_n$. The determinant is the product of its diagonal entries.
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What is the determinant of a triangular matrix in terms of its diagonal entries?
What is the determinant of a triangular matrix in terms of its diagonal entries?
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$\det(A)$ equals the product of the diagonal entries. For triangular matrices, the determinant is the product of the diagonal elements.
$\det(A)$ equals the product of the diagonal entries. For triangular matrices, the determinant is the product of the diagonal elements.
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What does it mean for the columns of $A$ if $A$ is invertible?
What does it mean for the columns of $A$ if $A$ is invertible?
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The columns are linearly independent. Invertibility requires the columns to form a linearly independent set.
The columns are linearly independent. Invertibility requires the columns to form a linearly independent set.
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What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?
What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?
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The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.
The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.
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What is the augmented matrix form of the system $A\vec{x}=\vec{b}$?
What is the augmented matrix form of the system $A\vec{x}=\vec{b}$?
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$[A\mid \vec{b}]$. The augmented matrix combines coefficients and constants for row reduction.
$[A\mid \vec{b}]$. The augmented matrix combines coefficients and constants for row reduction.
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What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?
What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?
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$A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.
$A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.
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Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?
Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?
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$\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.
$\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.
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What is $A^{-1}$ for $A=\begin{pmatrix}2&0\0&5\end{pmatrix}$?
What is $A^{-1}$ for $A=\begin{pmatrix}2&0\0&5\end{pmatrix}$?
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$A^{-1}=\begin{pmatrix}\frac{1}{2}&0\0&\frac{1}{5}\end{pmatrix}$. The inverse of a diagonal matrix reciprocates each diagonal entry.
$A^{-1}=\begin{pmatrix}\frac{1}{2}&0\0&\frac{1}{5}\end{pmatrix}$. The inverse of a diagonal matrix reciprocates each diagonal entry.
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What is $\det!\left(\begin{pmatrix}3&1\2&4\end{pmatrix}\right)$?
What is $\det!\left(\begin{pmatrix}3&1\2&4\end{pmatrix}\right)$?
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$10$. Computed using the 2x2 determinant formula $ad-bc$.
$10$. Computed using the 2x2 determinant formula $ad-bc$.
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Find $\left(\begin{pmatrix}1&2\3&4\end{pmatrix}\right)^{-1}$.
Find $\left(\begin{pmatrix}1&2\3&4\end{pmatrix}\right)^{-1}$.
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$\begin{pmatrix}-2&1\frac{3}{2}&-\frac{1}{2}\end{pmatrix}$. The 2x2 inverse formula scales the adjugate by $1/\det(A)$.
$\begin{pmatrix}-2&1\frac{3}{2}&-\frac{1}{2}\end{pmatrix}$. The 2x2 inverse formula scales the adjugate by $1/\det(A)$.
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Identify whether $A=\begin{pmatrix}1&2\2&4\end{pmatrix}$ is invertible.
Identify whether $A=\begin{pmatrix}1&2\2&4\end{pmatrix}$ is invertible.
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Not invertible because $\det(A)=0$. A zero determinant confirms the matrix is singular.
Not invertible because $\det(A)=0$. A zero determinant confirms the matrix is singular.
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