Matrices as Functions - AP Precalculus
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Define scalar multiplication of a matrix.
Define scalar multiplication of a matrix.
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Multiplying each element of a matrix by a scalar. Scale each entry by the scalar value.
Multiplying each element of a matrix by a scalar. Scale each entry by the scalar value.
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What is the associative property of matrix multiplication?
What is the associative property of matrix multiplication?
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$(AB)C = A(BC)$. Grouping doesn't affect the product.
$(AB)C = A(BC)$. Grouping doesn't affect the product.
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Is $\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$ antisymmetric?
Is $\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$ antisymmetric?
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Yes, it is antisymmetric. Check if matrix equals negative transpose.
Yes, it is antisymmetric. Check if matrix equals negative transpose.
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Define a square matrix.
Define a square matrix.
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A matrix with the same number of rows and columns. Equal dimensions make it square-shaped.
A matrix with the same number of rows and columns. Equal dimensions make it square-shaped.
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What is an antisymmetric matrix?
What is an antisymmetric matrix?
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A matrix where $A = -A^T$. Matrix equals negative of its transpose.
A matrix where $A = -A^T$. Matrix equals negative of its transpose.
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Is $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ symmetric?
Is $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ symmetric?
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Yes, it is symmetric. Check if matrix equals its transpose.
Yes, it is symmetric. Check if matrix equals its transpose.
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What is a symmetric matrix?
What is a symmetric matrix?
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A matrix where $A = A^T$. Matrix equals its own transpose.
A matrix where $A = A^T$. Matrix equals its own transpose.
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Find the trace of $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$.
Find the trace of $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$.
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The trace is $15$. Add diagonal elements: $4 + 5 + 6 = 15$.
The trace is $15$. Add diagonal elements: $4 + 5 + 6 = 15$.
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What is the trace of a matrix?
What is the trace of a matrix?
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The sum of the diagonal elements of a square matrix. Sum of main diagonal elements.
The sum of the diagonal elements of a square matrix. Sum of main diagonal elements.
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What is a diagonal matrix?
What is a diagonal matrix?
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A matrix where all off-diagonal elements are zero. Non-diagonal entries are all zero.
A matrix where all off-diagonal elements are zero. Non-diagonal entries are all zero.
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Find the determinant of $\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}$.
Find the determinant of $\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}$.
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The determinant is $7$. Calculate: $(3)(5) - (4)(2) = 15 - 8 = 7$.
The determinant is $7$. Calculate: $(3)(5) - (4)(2) = 15 - 8 = 7$.
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What condition makes a $2 \times 2$ matrix singular?
What condition makes a $2 \times 2$ matrix singular?
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The determinant is zero. Zero determinant means no inverse exists.
The determinant is zero. Zero determinant means no inverse exists.
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What is the determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?
What is the determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?
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$ad - bc$. Cross-multiply and subtract for 2×2.
$ad - bc$. Cross-multiply and subtract for 2×2.
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What is a matrix?
What is a matrix?
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A rectangular array of numbers arranged in rows and columns. Basic definition of a matrix structure.
A rectangular array of numbers arranged in rows and columns. Basic definition of a matrix structure.
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State the order of a matrix with 3 rows and 4 columns.
State the order of a matrix with 3 rows and 4 columns.
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The order is $3 \times 4$. Order is written as rows × columns.
The order is $3 \times 4$. Order is written as rows × columns.
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What is the element in the second row, third column of matrix $A$?
What is the element in the second row, third column of matrix $A$?
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$a_{2,3}$. Standard notation: $a_{row,column}$.
$a_{2,3}$. Standard notation: $a_{row,column}$.
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What is an invertible matrix?
What is an invertible matrix?
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A matrix with a non-zero determinant and an inverse. Has an inverse that undoes multiplication.
A matrix with a non-zero determinant and an inverse. Has an inverse that undoes multiplication.
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What is the commutative property of matrix multiplication?
What is the commutative property of matrix multiplication?
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Matrix multiplication is not commutative. Generally $AB ≠ BA$ for matrices.
Matrix multiplication is not commutative. Generally $AB ≠ BA$ for matrices.
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What is the associative property of matrix multiplication?
What is the associative property of matrix multiplication?
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$(AB)C = A(BC)$. Grouping doesn't affect the product.
$(AB)C = A(BC)$. Grouping doesn't affect the product.
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What is the identity property of matrix multiplication?
What is the identity property of matrix multiplication?
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$AI = IA = A$, where $I$ is the identity matrix. Identity matrix preserves multiplication.
$AI = IA = A$, where $I$ is the identity matrix. Identity matrix preserves multiplication.
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What condition is necessary for matrix multiplication $AB$?
What condition is necessary for matrix multiplication $AB$?
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The number of columns in $A$ must equal the number of rows in $B$. Inner dimensions must match for multiplication.
The number of columns in $A$ must equal the number of rows in $B$. Inner dimensions must match for multiplication.
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What is matrix multiplication?
What is matrix multiplication?
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The product of matrices by row-by-column multiplication. Dot product of rows and columns.
The product of matrices by row-by-column multiplication. Dot product of rows and columns.
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If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A - A$.
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A - A$.
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$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Subtracting gives the zero matrix.
$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Subtracting gives the zero matrix.
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What is the result of $A + 0$ for any matrix $A$?
What is the result of $A + 0$ for any matrix $A$?
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The matrix $A$ itself. Zero matrix is additive identity.
The matrix $A$ itself. Zero matrix is additive identity.
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What condition is necessary for two matrices to be added?
What condition is necessary for two matrices to be added?
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They must have the same dimensions. Matrices must be same size to add.
They must have the same dimensions. Matrices must be same size to add.
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What is matrix addition?
What is matrix addition?
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The sum of two matrices by adding corresponding elements. Add elements in same positions.
The sum of two matrices by adding corresponding elements. Add elements in same positions.
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What is the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?
What is the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?
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$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Formula for 2×2 matrix inverse.
$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Formula for 2×2 matrix inverse.
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If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, what is $2A$?
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, what is $2A$?
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$2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Multiply each element by the scalar 2.
$2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Multiply each element by the scalar 2.
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Define zero matrix.
Define zero matrix.
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A matrix in which all elements are zero. All entries equal zero.
A matrix in which all elements are zero. All entries equal zero.
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What is a column matrix?
What is a column matrix?
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A matrix with only one column. Single vertical array of elements.
A matrix with only one column. Single vertical array of elements.
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