Matrices And Determinants

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What is a matrix?

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A matrix is a rectangular array of numbers arranged in rows and columns.

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What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

What is a row matrix?

A matrix with only one row.

What is a column matrix?

A matrix with only one column.

What is a square matrix?

A matrix with the same number of rows and columns.

What is a diagonal matrix?

A square matrix where all elements outside the main diagonal are zero.

What is a scalar matrix?

A diagonal matrix where all the diagonal elements are the same.

What is an identity matrix?

A diagonal matrix where all the diagonal elements are 1.

What is a zero matrix?

A matrix where all elements are zero.

How can matrices be added or subtracted?

Matrices can be added or subtracted if they have the same dimensions, performed element-wise.

What is scalar multiplication of a matrix?

Each element of the matrix is multiplied by a scalar value.

What is the formula for matrix multiplication?

For matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the product is $C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$.

What is the transpose of a matrix?

The transpose of a matrix $A$ is obtained by swapping its rows with columns, denoted as $A^T$.

When does a matrix have an inverse?

A square matrix $A$ has an inverse $A^{-1}$ if and only if $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

What is the determinant of a $2 \times 2$ matrix?

For a matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $\text{det}(A) = ad - bc$.

What is the determinant of a $3 \times 3$ matrix?

For a matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$, the determinant is $\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$.

What is the multiplicative property of determinants?

$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$.

What happens when two rows of a matrix are swapped?

Swapping two rows changes the sign of the determinant.

What is Cramer's Rule?

Cramer's Rule is used to solve systems of linear equations if the determinant of the coefficient matrix is non-zero.

Matrices And Determinants

1 / 18

What is a matrix?

Flip

Spacebar

A matrix is a rectangular array of numbers arranged in rows and columns.

Flip

Spacebar

All flashcards

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

What is a row matrix?

A matrix with only one row.

What is a column matrix?

A matrix with only one column.

What is a square matrix?

A matrix with the same number of rows and columns.

What is a diagonal matrix?

A square matrix where all elements outside the main diagonal are zero.

What is a scalar matrix?

A diagonal matrix where all the diagonal elements are the same.

What is an identity matrix?

A diagonal matrix where all the diagonal elements are 1.

What is a zero matrix?

A matrix where all elements are zero.

How can matrices be added or subtracted?

Matrices can be added or subtracted if they have the same dimensions, performed element-wise.

What is scalar multiplication of a matrix?

Each element of the matrix is multiplied by a scalar value.

What is the formula for matrix multiplication?

For matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the product is $C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$.

What is the transpose of a matrix?

The transpose of a matrix $A$ is obtained by swapping its rows with columns, denoted as $A^T$.

When does a matrix have an inverse?

A square matrix $A$ has an inverse $A^{-1}$ if and only if $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

What is the determinant of a $2 \times 2$ matrix?

For a matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $\text{det}(A) = ad - bc$.

What is the determinant of a $3 \times 3$ matrix?

For a matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$, the determinant is $\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$.

What is the multiplicative property of determinants?

$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$.

What happens when two rows of a matrix are swapped?

Swapping two rows changes the sign of the determinant.

What is Cramer's Rule?

Cramer's Rule is used to solve systems of linear equations if the determinant of the coefficient matrix is non-zero.