Matrices - AP Precalculus
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What is a row matrix?
What is a row matrix?
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A matrix with a single row. Also called a row vector with dimension $1×n$.
A matrix with a single row. Also called a row vector with dimension $1×n$.
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What is the scalar multiplication of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ by 3?
What is the scalar multiplication of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ by 3?
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$\begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}$. Multiply each element by 3.
$\begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}$. Multiply each element by 3.
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What does it mean for two matrices to be equal?
What does it mean for two matrices to be equal?
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All corresponding entries are equal. Element-by-element equality for same dimensions.
All corresponding entries are equal. Element-by-element equality for same dimensions.
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Find the result of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.
Find the result of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.
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$\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Identity matrix preserves the second matrix unchanged.
$\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Identity matrix preserves the second matrix unchanged.
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What operation is performed by $2A$ for a matrix $A$?
What operation is performed by $2A$ for a matrix $A$?
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Scalar multiplication, doubling each element. Multiplies each matrix element by the scalar.
Scalar multiplication, doubling each element. Multiplies each matrix element by the scalar.
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Calculate the sum of matrices $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$.
Calculate the sum of matrices $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$.
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$\begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$. Add corresponding elements: $(1+4,2+3,3+2,4+1)$.
$\begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$. Add corresponding elements: $(1+4,2+3,3+2,4+1)$.
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Define the term 'symmetric matrix'.
Define the term 'symmetric matrix'.
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A matrix equal to its transpose, $A = A^T$. Matrix equals its own transpose.
A matrix equal to its transpose, $A = A^T$. Matrix equals its own transpose.
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Find the trace of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.
Find the trace of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.
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- Trace equals sum of diagonal elements: $1+4=5$.
- Trace equals sum of diagonal elements: $1+4=5$.
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What is a diagonal matrix?
What is a diagonal matrix?
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A matrix where non-diagonal elements are zero. Only diagonal entries can be non-zero.
A matrix where non-diagonal elements are zero. Only diagonal entries can be non-zero.
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Which property describes $AB = BA$ for matrices $A$ and $B$?
Which property describes $AB = BA$ for matrices $A$ and $B$?
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Commutative property (rarely holds for matrices). Matrix multiplication is generally not commutative.
Commutative property (rarely holds for matrices). Matrix multiplication is generally not commutative.
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What is the rank of a matrix?
What is the rank of a matrix?
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The maximum number of linearly independent row or column vectors. Measures the dimension of the row or column space.
The maximum number of linearly independent row or column vectors. Measures the dimension of the row or column space.
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Identify the inverse of a matrix $A$ if it exists.
Identify the inverse of a matrix $A$ if it exists.
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Matrix $A^{-1}$ such that $AA^{-1} = I$. Inverse satisfies the multiplicative identity property.
Matrix $A^{-1}$ such that $AA^{-1} = I$. Inverse satisfies the multiplicative identity property.
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What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?
What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?
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$ad - bc$. Formula for $2×2$ determinant calculation.
$ad - bc$. Formula for $2×2$ determinant calculation.
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What condition must two matrices meet to be added together?
What condition must two matrices meet to be added together?
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They must have the same dimensions. Addition is defined element-wise for matrices.
They must have the same dimensions. Addition is defined element-wise for matrices.
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What is the result of multiplying a matrix by the identity matrix?
What is the result of multiplying a matrix by the identity matrix?
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The original matrix. Identity matrix leaves any matrix unchanged when multiplied.
The original matrix. Identity matrix leaves any matrix unchanged when multiplied.
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How do you denote the transpose of a matrix $A$?
How do you denote the transpose of a matrix $A$?
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$A^T$. Standard notation for transpose operation.
$A^T$. Standard notation for transpose operation.
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Define an identity matrix.
Define an identity matrix.
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A square matrix with 1s on the main diagonal and 0s elsewhere. The multiplicative identity in matrix operations.
A square matrix with 1s on the main diagonal and 0s elsewhere. The multiplicative identity in matrix operations.
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What is a zero matrix?
What is a zero matrix?
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A matrix where all elements are zero. The additive identity in matrix operations.
A matrix where all elements are zero. The additive identity in matrix operations.
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Identify the main diagonal elements in a $3 \times 3$ matrix.
Identify the main diagonal elements in a $3 \times 3$ matrix.
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Elements $a_{11}$, $a_{22}$, $a_{33}$. Main diagonal runs from top-left to bottom-right.
Elements $a_{11}$, $a_{22}$, $a_{33}$. Main diagonal runs from top-left to bottom-right.
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What is a square matrix?
What is a square matrix?
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A matrix with the same number of rows and columns. Equal row and column counts define a square matrix.
A matrix with the same number of rows and columns. Equal row and column counts define a square matrix.
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How is the element in the second row, first column of a matrix denoted?
How is the element in the second row, first column of a matrix denoted?
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$a_{21}$. Subscript notation: first index is row, second is column.
$a_{21}$. Subscript notation: first index is row, second is column.
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State the dimensions of a $3 \times 4$ matrix.
State the dimensions of a $3 \times 4$ matrix.
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3 rows and 4 columns. Matrix dimensions are always written as rows × columns.
3 rows and 4 columns. Matrix dimensions are always written as rows × columns.
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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. The fundamental definition of a matrix structure.
A rectangular array of numbers arranged in rows and columns. The fundamental definition of a matrix structure.
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What is a column matrix?
What is a column matrix?
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A matrix with a single column. Also called a column vector with dimension $m×1$.
A matrix with a single column. Also called a column vector with dimension $m×1$.
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Check if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is invertible.
Check if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is invertible.
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Invertible, since determinant $\neq 0$. Determinant $= 1(4) - 2(3) = -2 ≠ 0$.
Invertible, since determinant $\neq 0$. Determinant $= 1(4) - 2(3) = -2 ≠ 0$.
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Determine the dimensions of zero matrix $O = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$.
Determine the dimensions of zero matrix $O = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$.
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$2 \times 3$. Matrix has 2 rows and 3 columns of zeros.
$2 \times 3$. Matrix has 2 rows and 3 columns of zeros.
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What is the inverse of the identity matrix $I$?
What is the inverse of the identity matrix $I$?
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The identity matrix $I$ itself. Identity matrix is its own inverse: $I^{-1} = I$.
The identity matrix $I$ itself. Identity matrix is its own inverse: $I^{-1} = I$.
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What is the result of transposing a column matrix?
What is the result of transposing a column matrix?
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A row matrix. Transpose flips dimensions from column to row.
A row matrix. Transpose flips dimensions from column to row.
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If $A$ is a $3 \times 2$ matrix, what are the dimensions of $A^T$?
If $A$ is a $3 \times 2$ matrix, what are the dimensions of $A^T$?
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$2 \times 3$. Transpose flips dimensions: rows become columns.
$2 \times 3$. Transpose flips dimensions: rows become columns.
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Can you multiply a $2 \times 3$ matrix by a $3 \times 2$ matrix?
Can you multiply a $2 \times 3$ matrix by a $3 \times 2$ matrix?
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Yes, resulting in a $2 \times 2$ matrix. Inner dimensions match (3), so multiplication is valid.
Yes, resulting in a $2 \times 2$ matrix. Inner dimensions match (3), so multiplication is valid.
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