Matrix Operations

Matrices are powerful tools for representing and solving mathematical problems. Before jumping into linear systems, like $A\vec{x} = \vec{b}$ , it's essential to understand how matrices behave and what operations we can perform on them.

Matrix Addition

You can add two matrices only if they have the same shape (same number of rows and columns).

Let:

A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

Then:

A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

Scalar Multiplication

You can also multiply a matrix by a scalar (single number):

k \cdot A = \begin{bmatrix} k a_{11} & k a_{12} \\ k a_{21} & k a_{22} \end{bmatrix}

Matrix Multiplication and Size Compatibility

Matrix multiplication is a row-by-column operation, not element-wise.

Rule: if matrix $A$ is of shape $(m \times n)$ and matrix $B$ is of shape $(n \times p)$ , then:

The multiplication $AB$ is valid;
The result will be a matrix of shape $(m \times p)$ .

Example:

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \ \ B = \begin{bmatrix} 5 \\ 6 \end{bmatrix}

$A$ is $(2 \times 2)$ and $B$ is $(2 \times 1)$ , then $AB$ is valid and results in a $(2 \times 1)$ matrix:

A \cdot B = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 6 \\ 3 \cdot 5 + 4 \cdot 6 \end{bmatrix} = \begin{bmatrix} 17 \\ 39 \end{bmatrix}

Transpose of a Matrix

The transpose of a matrix flips rows and columns. It is denoted as $A^T$ .

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Then:

A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Properties:

$(A^T)^T = A$ ;
$(A + B)^T = A^T + B^T$ ;
$(AB)^T = B^T A^T$ .

Determinant of a Matrix

2×2 Matrix

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

The determinant is:

\det(A) = ad - bc

3×3 Matrix

For:

A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

The determinant is:

\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This method is called cofactor expansion.

Larger matrices (4×4 and up) can be expanded recursively.
The determinant is useful because it indicates whether a matrix has an inverse (non-zero determinant).

Inverse of a Matrix

The inverse of a square matrix $A$ is denoted as $A^{-1}$ . It satisfies $A \cdot A^{-1} = I$ , where $I$ is the identity matrix.

Only square matrices with non-zero determinant have an inverse.

Example:

If matrix A is:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Then its inverse matrix $A^{-1}$ is:

A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Where $\det(A) \neq 0$ .

1. What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

2. What condition must be met for $AB$ to be valid?

3. What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

4. If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

5. Only square matrices can be multiplied.

6. The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .

What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

Select the correct answer

$(2 \times 2)$

$(2 \times 4)$

$(3 \times 3)$

$(3 \times 4)$

What condition must be met for $AB$ to be valid?

Select the correct answer

$A$ and $B$ must be square.

Same number of rows.

Columns of $A$ must match rows of $B$ .

Rows of $A$ must match columns of $B$ .

What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

Select the correct answer

$\begin{bmatrix}1 & 3 \\2 & 4\end{bmatrix}$

$\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$

$\begin{bmatrix}2 & 1 \\4 & 3\end{bmatrix}$

None of the above

If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

Select the correct answer

$(3 \times 3)$

$(3 \times 2)$

$(2 \times 3)$

Cannot be determined

Only square matrices can be multiplied.

Select the correct answer

True

False

The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .

Select the correct answer

True

False

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Matrix Operations

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Matrix Addition

You can add two matrices only if they have the same shape (same number of rows and columns).

Let:

A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

Then:

A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

Scalar Multiplication

You can also multiply a matrix by a scalar (single number):

k \cdot A = \begin{bmatrix} k a_{11} & k a_{12} \\ k a_{21} & k a_{22} \end{bmatrix}

Matrix Multiplication and Size Compatibility

Matrix multiplication is a row-by-column operation, not element-wise.

Rule: if matrix $A$ is of shape $(m \times n)$ and matrix $B$ is of shape $(n \times p)$ , then:

The multiplication $AB$ is valid;
The result will be a matrix of shape $(m \times p)$ .

Example:

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \ \ B = \begin{bmatrix} 5 \\ 6 \end{bmatrix}

$A$ is $(2 \times 2)$ and $B$ is $(2 \times 1)$ , then $AB$ is valid and results in a $(2 \times 1)$ matrix:

A \cdot B = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 6 \\ 3 \cdot 5 + 4 \cdot 6 \end{bmatrix} = \begin{bmatrix} 17 \\ 39 \end{bmatrix}

Transpose of a Matrix

The transpose of a matrix flips rows and columns. It is denoted as $A^T$ .

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Then:

A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Properties:

$(A^T)^T = A$ ;
$(A + B)^T = A^T + B^T$ ;
$(AB)^T = B^T A^T$ .

Determinant of a Matrix

2×2 Matrix

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

The determinant is:

\det(A) = ad - bc

3×3 Matrix

For:

A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

The determinant is:

\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This method is called cofactor expansion.

Larger matrices (4×4 and up) can be expanded recursively.
The determinant is useful because it indicates whether a matrix has an inverse (non-zero determinant).

Inverse of a Matrix

The inverse of a square matrix $A$ is denoted as $A^{-1}$ . It satisfies $A \cdot A^{-1} = I$ , where $I$ is the identity matrix.

Only square matrices with non-zero determinant have an inverse.

Example:

If matrix A is:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Then its inverse matrix $A^{-1}$ is:

A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Where $\det(A) \neq 0$ .

1. What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

2. What condition must be met for $AB$ to be valid?

3. What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

4. If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

5. Only square matrices can be multiplied.

6. The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .

What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

Select the correct answer

$(2 \times 2)$

$(2 \times 4)$

$(3 \times 3)$

$(3 \times 4)$

What condition must be met for $AB$ to be valid?

Select the correct answer

$A$ and $B$ must be square.

Same number of rows.

Columns of $A$ must match rows of $B$ .

Rows of $A$ must match columns of $B$ .

What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

Select the correct answer

$\begin{bmatrix}1 & 3 \\2 & 4\end{bmatrix}$

$\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$

$\begin{bmatrix}2 & 1 \\4 & 3\end{bmatrix}$

None of the above

If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

Select the correct answer

$(3 \times 3)$

$(3 \times 2)$

$(2 \times 3)$

Cannot be determined

Only square matrices can be multiplied.

Select the correct answer

True

False

The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .

Select the correct answer

True

False

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Секція 4. Розділ 3

Matrix Operations

Matrix Addition

Scalar Multiplication

Matrix Multiplication and Size Compatibility

Transpose of a Matrix

Determinant of a Matrix

2×2 Matrix

3×3 Matrix

Inverse of a Matrix

1. What is the shape of the product of a (2×3)(2 \times 3)(2×3) matrix and a (3×4)(3 \times 4)(3×4) matrix?

2. What condition must be met for ABABAB to be valid?

3. What is the transpose of [1234]\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}[13​24​] matrix?

4. If AAA is a (3×3)(3 \times 3)(3×3) matrix and BBB is (3×2)(3 \times 2)(3×2), what is the shape of ABABAB?

5. Only square matrices can be multiplied.

6. The transpose of a (2×3)(2 \times 3)(2×3) matrix is (3×2)(3 \times 2)(3×2).

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Matrix Operations

Matrix Addition

Scalar Multiplication

Matrix Multiplication and Size Compatibility

Transpose of a Matrix

Determinant of a Matrix

2×2 Matrix

3×3 Matrix

Inverse of a Matrix

1. What is the shape of the product of a (2×3)(2 \times 3)(2×3) matrix and a (3×4)(3 \times 4)(3×4) matrix?

2. What condition must be met for ABABAB to be valid?

3. What is the transpose of [1234]\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}[13​24​] matrix?

4. If AAA is a (3×3)(3 \times 3)(3×3) matrix and BBB is (3×2)(3 \times 2)(3×2), what is the shape of ABABAB?

5. Only square matrices can be multiplied.

6. The transpose of a (2×3)(2 \times 3)(2×3) matrix is (3×2)(3 \times 2)(3×2).

1. What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

2. What condition must be met for $AB$ to be valid?

3. What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

4. If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

6. The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .

1. What is the shape of the product of a $(2 \times 3)$ matrix and a $(3 \times 4)$ matrix?

2. What condition must be met for $AB$ to be valid?

3. What is the transpose of $\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}$ matrix?

4. If $A$ is a $(3 \times 3)$ matrix and $B$ is $(3 \times 2)$ , what is the shape of $AB$ ?

6. The transpose of a $(2 \times 3)$ matrix is $(3 \times 2)$ .