Zero and Identity Matrices and Determinants - Pre-Calculus
Card 1 of 30
Compute $\det!\left(\begin{pmatrix}3&1\2&1\end{pmatrix}\right)$.
Compute $\det!\left(\begin{pmatrix}3&1\2&1\end{pmatrix}\right)$.
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$1$. $3(1)-1(2)=3-2=1$; matrix is invertible.
$1$. $3(1)-1(2)=3-2=1$; matrix is invertible.
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Compute $\det!\left(\begin{pmatrix}2&1\4&2\end{pmatrix}\right)$.
Compute $\det!\left(\begin{pmatrix}2&1\4&2\end{pmatrix}\right)$.
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$0$. $2(2)-1(4)=4-4=0$; rows are proportional.
$0$. $2(2)-1(4)=4-4=0$; rows are proportional.
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If $\det(A)=0$ for a square matrix $A$, what can you conclude about $A^{-1}$?
If $\det(A)=0$ for a square matrix $A$, what can you conclude 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 a square matrix $A$ has an inverse, what must be true about $\det(A)$?
If a square matrix $A$ has an inverse, what must be true about $\det(A)$?
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$\det(A) \ne 0$. Invertible matrices must have nonzero determinant.
$\det(A) \ne 0$. Invertible matrices must have nonzero determinant.
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What is the determinant formula for $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the determinant formula for $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$\det(A)=ad-bc$. Standard formula for $2 \times 2$ determinant.
$\det(A)=ad-bc$. Standard formula for $2 \times 2$ determinant.
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What is $\det(I_n)$ for any positive integer $n$?
What is $\det(I_n)$ for any positive integer $n$?
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$1$. Identity matrix always has determinant 1.
$1$. Identity matrix always has determinant 1.
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What is the value of $0_{m \times n}B$ when $B$ is $n \times p$ and the product is defined?
What is the value of $0_{m \times n}B$ when $B$ is $n \times p$ and the product is defined?
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$0_{m \times p}$. Zero matrix times any matrix gives zero matrix.
$0_{m \times p}$. Zero matrix times any matrix gives zero matrix.
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What is the value of $A0_{n \times p}$ when $A$ is $m \times n$ and the product is defined?
What is the value of $A0_{n \times p}$ when $A$ is $m \times n$ and the product is defined?
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$0_{m \times p}$. Any matrix times zero matrix gives zero matrix.
$0_{m \times p}$. Any matrix times zero matrix gives zero matrix.
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What is the value of $I_mA$ when $A$ is an $m \times n$ matrix?
What is the value of $I_mA$ when $A$ is an $m \times n$ matrix?
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$A$. Left multiplication by identity preserves the matrix.
$A$. Left multiplication by identity preserves the matrix.
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What is the value of $AI_n$ when $A$ is an $m \times n$ matrix?
What is the value of $AI_n$ when $A$ is an $m \times n$ matrix?
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$A$. Right multiplication by identity preserves the matrix.
$A$. Right multiplication by identity preserves the matrix.
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What is the value of $0_{m \times n} + A$ for any $m \times n$ matrix $A$?
What is the value of $0_{m \times n} + A$ for any $m \times n$ matrix $A$?
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$A$. Addition is commutative; zero matrix leaves $A$ unchanged.
$A$. Addition is commutative; zero matrix leaves $A$ unchanged.
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What is the value of $A + 0_{m \times n}$ for any $m \times n$ matrix $A$?
What is the value of $A + 0_{m \times n}$ for any $m \times n$ matrix $A$?
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$A$. Zero matrix is the additive identity.
$A$. Zero matrix is the additive identity.
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What is the standard form of the $3 \times 3$ identity matrix $I_3$?
What is the standard form of the $3 \times 3$ identity matrix $I_3$?
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$\begin{pmatrix}1&0&0\0&1&0\0&0&1\end{pmatrix}$. Diagonal entries are 1, all other entries are 0.
$\begin{pmatrix}1&0&0\0&1&0\0&0&1\end{pmatrix}$. Diagonal entries are 1, all other entries are 0.
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What is the standard form of the $2 \times 2$ identity matrix $I_2$?
What is the standard form of the $2 \times 2$ identity matrix $I_2$?
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$\begin{pmatrix}1&0\0&1\end{pmatrix}$. Diagonal entries are 1, off-diagonal entries are 0.
$\begin{pmatrix}1&0\0&1\end{pmatrix}$. Diagonal entries are 1, off-diagonal entries are 0.
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What equation defines the multiplicative identity property for square matrices $A$ and $I$?
What equation defines the multiplicative identity property for square matrices $A$ and $I$?
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$AI = IA = A$. Multiplying by identity leaves square matrices unchanged.
$AI = IA = A$. Multiplying by identity leaves square matrices unchanged.
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What is the multiplicative identity matrix for $n \times n$ matrices?
What is the multiplicative identity matrix for $n \times n$ matrices?
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The identity matrix $I_n$. Has 1s on diagonal, 0s elsewhere; acts like 1 in multiplication.
The identity matrix $I_n$. Has 1s on diagonal, 0s elsewhere; acts like 1 in multiplication.
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What equation defines the additive identity property for matrices $A$ and $0$?
What equation defines the additive identity property for matrices $A$ and $0$?
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$A + 0 = A$. Adding zero matrix leaves any matrix unchanged.
$A + 0 = A$. Adding zero matrix leaves any matrix unchanged.
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What is $A^0$ if $A$ is any $4 \times 2$ matrix and $0$ is the $2 \times 3$ zero matrix?
What is $A^0$ if $A$ is any $4 \times 2$ matrix and $0$ is the $2 \times 3$ zero matrix?
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$A^0 = 0$ (the $4 \times 3$ zero matrix). Product dimensions are $4 \times 3$, all entries zero.
$A^0 = 0$ (the $4 \times 3$ zero matrix). Product dimensions are $4 \times 3$, all entries zero.
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What is the inverse of the identity matrix $I_n$?
What is the inverse of the identity matrix $I_n$?
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$I_n^{-1} = I_n$. Identity matrix is its own inverse: $I_n \cdot I_n = I_n$.
$I_n^{-1} = I_n$. Identity matrix is its own inverse: $I_n \cdot I_n = I_n$.
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What is the $m \times n$ zero matrix, and what entries does it contain?
What is the $m \times n$ zero matrix, and what entries does it contain?
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The $m \times n$ matrix with every entry equal to $0$. All entries are zero, regardless of matrix dimensions.
The $m \times n$ matrix with every entry equal to $0$. All entries are zero, regardless of matrix dimensions.
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What is the $n \times n$ identity matrix $I_n$ in terms of its entries?
What is the $n \times n$ identity matrix $I_n$ in terms of its entries?
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$I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal entries are 1, creating the multiplicative identity.
$I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal entries are 1, creating the multiplicative identity.
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What is the additive identity property for matrices using the zero matrix $0$?
What is the additive identity property for matrices using the zero matrix $0$?
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$A + 0 = A$ and $0 + A = A$. Zero matrix acts like 0 in real numbers for addition.
$A + 0 = A$ and $0 + A = A$. Zero matrix acts like 0 in real numbers for addition.
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What is the multiplicative identity property for square matrices using $I_n$?
What is the multiplicative identity property for square matrices using $I_n$?
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If $A$ is $n \times n$, then $AI_n = A$ and $I_nA = A$. Identity matrix preserves any square matrix under multiplication.
If $A$ is $n \times n$, then $AI_n = A$ and $I_nA = A$. Identity matrix preserves any square matrix under multiplication.
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What is the result of multiplying any matrix $A$ by a compatible zero matrix $0$?
What is the result of multiplying any matrix $A$ by a compatible zero matrix $0$?
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$A^0 = 0$ and $0A = 0$ (when the products are defined). Any matrix times zero matrix yields zero matrix.
$A^0 = 0$ and $0A = 0$ (when the products are defined). Any matrix times zero matrix yields zero matrix.
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What size must the identity matrix be so that $AI = A$ for an $m \times n$ matrix $A$?
What size must the identity matrix be so that $AI = A$ for an $m \times n$ matrix $A$?
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Use $I_n$ on the right: $A I_n = A$. Right multiplication requires $n \times n$ identity for $m \times n$ matrix.
Use $I_n$ on the right: $A I_n = A$. Right multiplication requires $n \times n$ identity for $m \times n$ matrix.
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What size must the identity matrix be so that $IA = A$ for an $m \times n$ matrix $A$?
What size must the identity matrix be so that $IA = A$ for an $m \times n$ matrix $A$?
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Use $I_m$ on the left: $I_m A = A$. Left multiplication requires $m \times m$ identity for $m \times n$ matrix.
Use $I_m$ on the left: $I_m A = A$. Left multiplication requires $m \times m$ identity for $m \times n$ matrix.
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What is the definition of an inverse matrix $A^{-1}$ for an $n \times n$ matrix $A$?
What is the definition of an inverse matrix $A^{-1}$ for an $n \times n$ matrix $A$?
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$AA^{-1} = I_n$ and $A^{-1}A = I_n$. Inverse matrix multiplied by original yields identity.
$AA^{-1} = I_n$ and $A^{-1}A = I_n$. Inverse matrix multiplied by original yields identity.
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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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$A$ is invertible iff there exists $A^{-1}$ with $AA^{-1}=A^{-1}A=I_n$. Matrix has inverse that yields identity when multiplied.
$A$ is invertible iff there exists $A^{-1}$ with $AA^{-1}=A^{-1}A=I_n$. Matrix has inverse that yields identity when multiplied.
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What determinant condition is equivalent to an $n \times n$ matrix $A$ being invertible?
What determinant condition is equivalent to an $n \times n$ matrix $A$ being invertible?
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$A$ is invertible iff $\det(A) \ne 0$. Nonzero determinant guarantees matrix has inverse.
$A$ is invertible iff $\det(A) \ne 0$. Nonzero determinant guarantees matrix has inverse.
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What conclusion can you make if $\det(A) = 0$ for a square matrix $A$?
What conclusion can you make if $\det(A) = 0$ for a square matrix $A$?
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$A$ is not invertible (it is singular). Zero determinant means no inverse exists.
$A$ is not invertible (it is singular). Zero determinant means no inverse exists.
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