Zero and Identity Matrices and Determinants - Geometry
Card 1 of 30
Compute $\det\left(\begin{pmatrix}2&0&0\0&-1&0\0&0&4\end{pmatrix}\right)$.
Compute $\det\left(\begin{pmatrix}2&0&0\0&-1&0\0&0&4\end{pmatrix}\right)$.
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$-8$. Diagonal matrix: product $2 \times (-1) \times 4$.
$-8$. Diagonal matrix: product $2 \times (-1) \times 4$.
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If $\det(A)=3$ and $\det(B)=5$ for $2\times 2$ matrices, what is $\det(AB)$?
If $\det(A)=3$ and $\det(B)=5$ for $2\times 2$ matrices, what is $\det(AB)$?
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$15$. Using multiplicative property: $3 \times 5 = 15$.
$15$. Using multiplicative property: $3 \times 5 = 15$.
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What is $\det(0_n)$ for any positive integer $n$?
What is $\det(0_n)$ for any positive integer $n$?
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$\det(0_n)=0$. Zero matrix has zero determinant always.
$\det(0_n)=0$. Zero matrix has zero determinant always.
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What is the $n\times n$ identity matrix, written $I_n$?
What is the $n\times n$ identity matrix, written $I_n$?
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$I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal of ones makes it the multiplicative identity.
$I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal of ones makes it the multiplicative identity.
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What is the additive identity property for matrices?
What is the additive identity property for matrices?
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For any $A$, $A+0=A$. Zero matrix acts like zero in addition.
For any $A$, $A+0=A$. Zero matrix acts like zero in addition.
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If $A$ is invertible, what is $\det(A^{-1})$ in terms of $\det(A)$?
If $A$ is invertible, what is $\det(A^{-1})$ in terms of $\det(A)$?
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$\det(A^{-1})=\frac{1}{\det(A)}$. Inverse determinant is reciprocal of original.
$\det(A^{-1})=\frac{1}{\det(A)}$. Inverse determinant is reciprocal of original.
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Which matrix is the additive identity for $2\times 2$ matrices?
Which matrix is the additive identity for $2\times 2$ matrices?
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$\begin{pmatrix}0&0\0&0\end{pmatrix}$. Standard form of $2 \times 2$ zero matrix.
$\begin{pmatrix}0&0\0&0\end{pmatrix}$. Standard form of $2 \times 2$ zero matrix.
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What is $A+I_n$ called when $A$ is $n\times n$?
What is $A+I_n$ called when $A$ is $n\times n$?
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It is the matrix sum of $A$ and $I_n$ (defined entrywise). Matrix addition performed entrywise on compatible matrices.
It is the matrix sum of $A$ and $I_n$ (defined entrywise). Matrix addition performed entrywise on compatible matrices.
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Identify the correct statement about $I_n$ and $0_n$ under multiplication.
Identify the correct statement about $I_n$ and $0_n$ under multiplication.
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$AI_n=A$ and $A0_n=0_n$ (for $n\times n$ matrix $A$). Identity acts as 1, zero acts as 0.
$AI_n=A$ and $A0_n=0_n$ (for $n\times n$ matrix $A$). Identity acts as 1, zero acts as 0.
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Which matrix is the multiplicative identity for $2\times 2$ matrices?
Which matrix is the multiplicative identity for $2\times 2$ matrices?
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$\begin{pmatrix}1&0\0&1\end{pmatrix}$. Standard form of $2 \times 2$ identity matrix.
$\begin{pmatrix}1&0\0&1\end{pmatrix}$. Standard form of $2 \times 2$ identity matrix.
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If $A$ is invertible, what must be true about $\det(A)$?
If $A$ is invertible, what must be true about $\det(A)$?
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$\det(A)\ne 0$. Invertible matrices have nonzero determinants.
$\det(A)\ne 0$. Invertible matrices have nonzero determinants.
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If $\det(A)=0$, what is the correct conclusion about $A^{-1}$?
If $\det(A)=0$, what is the correct conclusion about $A^{-1}$?
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$A^{-1}$ does not exist. Zero determinant means matrix is not invertible.
$A^{-1}$ does not exist. Zero determinant means matrix is not invertible.
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If $\det(A)=\frac{1}{4}$, what is $\det(A^{-1})$?
If $\det(A)=\frac{1}{4}$, what is $\det(A^{-1})$?
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$4$. Inverse determinant is reciprocal: $\frac{1}{1/4} = 4$.
$4$. Inverse determinant is reciprocal: $\frac{1}{1/4} = 4$.
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If $\det(A)=-2$ for a $3\times 3$ matrix, what is $\det(2A)$?
If $\det(A)=-2$ for a $3\times 3$ matrix, what is $\det(2A)$?
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$2^3\det(A)=-16$. Scalar multiplication: $2^3 \times (-2) = -16$.
$2^3\det(A)=-16$. Scalar multiplication: $2^3 \times (-2) = -16$.
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If $\det(A)=3$ and $\det(B)=5$ for $2\times 2$ matrices, what is $\det(AB)$?
If $\det(A)=3$ and $\det(B)=5$ for $2\times 2$ matrices, what is $\det(AB)$?
Tap to reveal answer
$15$. Using multiplicative property: $3 \times 5 = 15$.
$15$. Using multiplicative property: $3 \times 5 = 15$.
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Compute $\det\left(\begin{pmatrix}2&0&0\0&-1&0\0&0&4\end{pmatrix}\right)$.
Compute $\det\left(\begin{pmatrix}2&0&0\0&-1&0\0&0&4\end{pmatrix}\right)$.
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$-8$. Diagonal matrix: product $2 \times (-1) \times 4$.
$-8$. Diagonal matrix: product $2 \times (-1) \times 4$.
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What is the inverse formula for $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ when $\det(A)\ne 0$?
What is the inverse formula for $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ when $\det(A)\ne 0$?
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$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. Derived from determinant formula and cofactor method.
$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. Derived from determinant formula and cofactor method.
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Which determinant value guarantees that a $2\times 2$ matrix has no inverse?
Which determinant value guarantees that a $2\times 2$ matrix has no inverse?
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$\det(A)=0$. Zero determinant prevents inverse from existing.
$\det(A)=0$. Zero determinant prevents inverse from existing.
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If $\det(A)\ne 0$, what must be true about the equation $AX=I$?
If $\det(A)\ne 0$, what must be true about the equation $AX=I$?
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It has the unique solution $X=A^{-1}$. Nonzero determinant guarantees unique solution exists.
It has the unique solution $X=A^{-1}$. Nonzero determinant guarantees unique solution exists.
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What is the determinant of a product of square matrices $A$ and $B$ of the same size?
What is the determinant of a product of square matrices $A$ and $B$ of the same size?
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$\det(AB)=\det(A)\det(B)$. Determinant is multiplicative for square matrices.
$\det(AB)=\det(A)\det(B)$. Determinant is multiplicative for square matrices.
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What is the additive inverse property for matrices?
What is the additive inverse property for matrices?
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For any $A$, $A+(-A)=0$. Adding a matrix to its negative gives zero.
For any $A$, $A+(-A)=0$. Adding a matrix to its negative gives zero.
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What is the multiplicative identity property for square matrices?
What is the multiplicative identity property for square matrices?
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For any $n\times n$ matrix $A$, $AI_n=I_nA=A$. Identity matrix acts like 1 in multiplication.
For any $n\times n$ matrix $A$, $AI_n=I_nA=A$. Identity matrix acts like 1 in multiplication.
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What is the result of multiplying any matrix $A$ by a compatible zero matrix?
What is the result of multiplying any matrix $A$ by a compatible zero matrix?
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$A^0=0$ and $0A=0$ (with compatible sizes). Zero matrix annihilates in multiplication.
$A^0=0$ and $0A=0$ (with compatible sizes). Zero matrix annihilates in multiplication.
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What does it mean for a square matrix $A$ to be invertible?
What does it mean for a square matrix $A$ to be invertible?
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There exists $A^{-1}$ such that $AA^{-1}=A^{-1}A=I$. The inverse undoes multiplication to give identity.
There exists $A^{-1}$ such that $AA^{-1}=A^{-1}A=I$. The inverse undoes multiplication to give identity.
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What is the key determinant test for invertibility of a square matrix $A$?
What is the key determinant test for invertibility of a square matrix $A$?
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$A$ is invertible if and only if $\det(A)\ne 0$. Nonzero determinant is the invertibility criterion.
$A$ is invertible if and only if $\det(A)\ne 0$. Nonzero determinant is the invertibility criterion.
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Identify the additive identity in the set of all $m\times n$ matrices.
Identify the additive identity in the set of all $m\times n$ matrices.
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The $m\times n$ zero matrix $0$. Zero matrix is unique additive neutral element.
The $m\times n$ zero matrix $0$. Zero matrix is unique additive neutral element.
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Identify the multiplicative identity in the set of all $n\times n$ matrices.
Identify the multiplicative identity in the set of all $n\times n$ matrices.
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The $n\times n$ identity matrix $I_n$. Identity matrix is unique multiplicative neutral element.
The $n\times n$ identity matrix $I_n$. Identity matrix is unique multiplicative neutral element.
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What is $A+0_{m\times n}$ for an $m\times n$ matrix $A$?
What is $A+0_{m\times n}$ for an $m\times n$ matrix $A$?
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$A+0_{m\times n}=A$. Adding zero matrix leaves $A$ unchanged.
$A+0_{m\times n}=A$. Adding zero matrix leaves $A$ unchanged.
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What is $A- A$ for any matrix $A$?
What is $A- A$ for any matrix $A$?
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$A-A=0$. Subtracting a matrix from itself gives zero.
$A-A=0$. Subtracting a matrix from itself gives zero.
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What is $A\cdot 0_{n\times p}$ when $A$ is $m\times n$?
What is $A\cdot 0_{n\times p}$ when $A$ is $m\times n$?
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$A0_{n\times p}=0_{m\times p}$. Multiplying by zero matrix gives zero result.
$A0_{n\times p}=0_{m\times p}$. Multiplying by zero matrix gives zero result.
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