Determinants - Determinants notations and outline
- Definition of a determinant
- Notation used for determinants
- Outline of the topic
Definition of a determinant
- A determinant is a scalar value that can be computed from the elements of a square matrix.
- It is used to determine various properties of the matrix, such as invertibility and solutions to linear equations.
Example:
Consider the matrix A:
[
A =
\begin{bmatrix}
3 & 1 \
\end{bmatrix}
]
The determinant of matrix A, denoted as |A| or det(A), is calculated as follows:
[
|A| = (2 \times 1) - (5 \times 3) = -13
]
Notation used for determinants
- Determinants are commonly denoted using vertical bars or brackets, such as |A| or [A].
- The vertical bar notation is more commonly used in mathematics.
- The size of the matrix is represented as a subscript within the notation, such as |A| for a 2x2 matrix.
Example:
For a 3x3 matrix B:
[
B =
\begin{bmatrix}
7 & 8 & 9 \
\end{bmatrix}
]
The determinant of matrix B can be written as |B| or det(B).
- Introduction to determinants.
- Methods for calculating determinants of 2x2 matrices.
- Direct multiplication method.
- Cross multiplication method.
- Determinants of 3x3 matrices.
- Expansion along rows or columns.
- Cofactor expansion method.
- Properties of determinants.
- Multiplicative property.
- Transpose property.
- Row or column scaling property.
- Applications of determinants.
- Solving systems of linear equations.
- Finding the inverse of a matrix.
- Higher order determinants.
- Cofactor expansion method for larger matrices.
- Laplace expansion method.
Example:
Solve the following system of linear equations using determinants:
[
\begin{align*}
2x + 3y &= 7 \
5x - 2y &= -8 \
\end{align*}
]
By using determinants, we can write the equations in matrix form as:
[
\begin{bmatrix}
2 & 3 \
5 & -2 \
\end{bmatrix}
\begin{bmatrix}
x \
y \
\end{bmatrix}
=
\begin{bmatrix}
7 \
-8 \
\end{bmatrix}
]
Then, we can use determinants to find the values of x and y.
Note: Continue the lecture by discussing each point in detail and providing additional examples and equations where relevant.
Determinants - Determinants notations and outline
Methods for calculating determinants of 2x2 matrices
- Direct multiplication method:
- Multiply the top left element by the bottom right element.
- Subtract the product of the top right and bottom left elements.
- Cross multiplication method:
- Multiply the top left element by the bottom right element.
- Multiply the top right element by the bottom left element.
- Subtract the results of the two multiplications.
Example:
Consider the matrix C:
[
C =
\begin{bmatrix}
4 & 5 \
\end{bmatrix}
]
To calculate the determinant of matrix C using the direct multiplication method:
[
|C| = (2 \times 5) - (3 \times 4) = -2
]
Determinants of 3x3 matrices
- Expansion along rows or columns:
- Choose a row or column to expand.
- Multiply each element in that row or column by its minor determinant, obtained by eliminating the row and column it belongs to.
- Alternate the signs of the products and sum them up.
- Cofactor expansion method:
- Choose a row or column to expand.
- Multiply each element in that row or column by its cofactor, obtained by multiplying its minor determinant by (-1) raised to the power of its row and column indices.
- Sum up the products.
Example:
Consider the matrix D:
[
D =
\begin{bmatrix}
7 & 8 & 9 \
\end{bmatrix}
]
To calculate the determinant of matrix D using the expansion along rows or columns:
[
|D| = (1 \times |E|) - (2 \times |F|) + (3 \times |G|)
]
Where E, F, G are the 2x2 matrices obtained by eliminating rows and columns from matrix D.
Properties of determinants
- Multiplicative property:
- The determinant of a product of two matrices is equal to the product of their determinants.
- Transpose property:
- The determinant of a matrix is equal to the determinant of its transpose.
- Row or column scaling property:
- If a row or column of a matrix is multiplied by a scalar k, the determinant is multiplied by k.
Example:
Consider the matrix A:
[
A =
\begin{bmatrix}
3 & 1 \
\end{bmatrix}
]
The determinant of matrix A is -13.
For matrix B = 2A, the determinant of B will be -26 due to the scaling property.
Applications of determinants
- Solving systems of linear equations:
- If the determinant of the coefficient matrix is non-zero, the system has a unique solution.
- If the determinant is zero, the system may have infinitely many solutions or no solution.
- Finding the inverse of a matrix:
- A square matrix is invertible if its determinant is non-zero.
- The inverse of a matrix can be found using the formula: [A^{-1} = \frac{1}{|A|} \times \text{adj}(A)]
Example:
Consider the system of linear equations:
[
\begin{align*}
2x + 3y &= 7 \
5x - 2y &= -8 \
\end{align*}
]
Using determinants, we can determine if the system has a unique solution or not.
Higher order determinants
- Cofactor expansion method for larger matrices:
- Choose a row or column to expand.
- Multiply each element in that row or column by its cofactor, obtained by multiplying its minor determinant by (-1) raised to the power of its row and column indices.
- Sum up the products.
- Laplace expansion method:
- Choose any row or column to expand.
- Multiply each element in that row or column by its cofactor.
- Sum up the products.
Example:
Consider the matrix E:
[
E =
\begin{bmatrix}
7 & 8 & 9 \
\end{bmatrix}
]
To calculate the determinant of matrix E using the Laplace expansion method:
[
|E| = 1 \times C_{11} + 2 \times C_{12} + 3 \times C_{13}
]
Where C_{ij} represents the cofactor of the element in the i-th row and j-th column.
Determinants - Determinants notations and outline
- Definition of a determinant
- Notation used for determinants
- Outline of the topic
Definition of a determinant
- A determinant is a scalar value that can be computed from the elements of a square matrix.
- It is used to determine various properties of the matrix, such as invertibility and solutions to linear equations.
Example:
Consider the matrix A:
[
A =
\begin{bmatrix}
3 & 1 \
\end{bmatrix}
]
The determinant of matrix A, denoted as |A| or det(A), is calculated as follows:
[
|A| = (2 \times 1) - (5 \times 3) = -13
]
Notation used for determinants
- Determinants are commonly denoted using vertical bars or brackets, such as |A| or [A].
- The vertical bar notation is more commonly used in mathematics.
- The size of the matrix is represented as a subscript within the notation, such as |A| for a 2x2 matrix.
Example:
For a 3x3 matrix B:
[
B =
\begin{bmatrix}
7 & 8 & 9 \
\end{bmatrix}
]
The determinant of matrix B can be written as |B| or det(B).
- Introduction to determinants.
- Methods for calculating determinants of 2x2 matrices.
- Direct multiplication method.
- Cross multiplication method.
- Determinants of 3x3 matrices.
- Expansion along rows or columns.
- Cofactor expansion method.
- Properties of determinants.
- Multiplicative property.
- Transpose property.
- Row or column scaling property.
- Applications of determinants.
- Solving systems of linear equations.
- Finding the inverse of a matrix.
- Higher order determinants.
- Cofactor expansion method for larger matrices.
- Laplace expansion method.
Example:
Solve the following system of linear equations using determinants:
[
\begin{align*}
2x + 3y &= 7 \
5x - 2y &= -8 \
\end{align*}
]
By using determinants, we can write the equations in matrix form as:
[
\begin{bmatrix}
2 & 3 \
5 & -2 \
\end{bmatrix}
\begin{bmatrix}
x \
y \
\end{bmatrix}
=
\begin{bmatrix}
7 \
-8 \
\end{bmatrix}
]
Then, we can use determinants to find the values of x and y.
Thank you for your attention
- Feel free to ask any questions you may have.
- Next lecture: Solving systems of linear equations using determinants.
Sure! Here are slides 21-30:
Methods for calculating determinants of 2x2 matrices
- Direct multiplication method:
- Multiply the top left element by the bottom right element.
- Subtract the product of the top right and bottom left elements.
- Cross multiplication method:
- Multiply the top left element by the bottom right element.
- Multiply the top right element by the bottom left element.
- Subtract the results of the two multiplications.
Example:
Consider the matrix C:
C = [2 3] [4 5]
To calculate the determinant of matrix C using the direct multiplication method:
|C| = (2 * 5) - (3 * 4) = -2
Determinants of 3x3 matrices
- Expansion along rows or columns:
- Choose a row or column to expand.
- Multiply each element in that row or column by its minor determinant, obtained by eliminating the row and column it belongs to.
- Alternate the signs of the products and sum them up.
- Cofactor expansion method:
- Choose a row or column to expand.
- Multiply each element in that row or column by its cofactor, obtained by multiplying its minor determinant by (-1) raised to the power of its row and column indices.
- Sum up the products.
Example:
Consider the matrix D:
D = [1 2 3] [4 5 6] [7 8 9]
To calculate the determinant of matrix D using the expansion along rows or columns:
|D| = (1 * |E|) - (2 * |F|) + (3 * |G|)
Where E, F, G are the 2x2 matrices obtained by eliminating rows and columns from matrix D.
Properties of determinants
- Multiplicative property:
- The determinant of a product of two matrices is equal to the product of their determinants.
- Transpose property:
- The determinant of a matrix is equal to the determinant of its transpose.
- Row or column scaling property:
- If a row or column of a matrix is multiplied by a scalar k, the determinant is multiplied by k.
Example:
Consider the matrix A:
A = [2 5] [3 1]
The determinant of matrix A is -13.
For matrix B = 2A, the determinant of B will be -26 due to the scaling property.
Applications of determinants
- Solving systems of linear equations:
- If the determinant of the coefficient matrix is non-zero, the system has a unique solution.
- If the determinant is zero, the system may have infinitely many solutions or no solution.
- Finding the inverse of a matrix:
- A square matrix is invertible if its determinant is non-zero.
- The inverse of a matrix can be found using the formula: A_inverse = (1/|A|) * adj(A)
Example:
Consider the system of linear equations:
``
5x - 2y = -8
``
Using determinants, we can determine if the system has a unique solution or not.
Higher order determinants
- Cofactor expansion method for larger matrices:
- Choose a row or column to expand.
- Multiply each element in that row or column by its cofactor, obtained by multiplying its minor determinant by (-1) raised to the power of its row and column indices.
- Sum up the products.
- Laplace expansion method:
- Choose any row or column to expand.
- Multiply each element in that row or column by its cofactor.
- Sum up the products.
Example:
Consider the matrix E:
E = [1 2 3] [4 5 6] [7 8 9]
To calculate the determinant of matrix E using the Laplace expansion method:
|E| = 1 * C_(1,1) + 2 * C_(1,2) + 3 * C_(1,3)
Where C_(i,j) represents the cofactor of the element in the i-th row and j-th column.
Determinants - Determinants notations and outline
- Definition of a determinant
- Notation used for determinants
- Outline of the topic
Definition of a determinant
- A determinant is a scalar value that can be computed from the elements of a square matrix.
- It is used to determine various properties of the matrix, such as invertibility and solutions to linear equations.
Example:
Consider the matrix A:
A = [2 5] [3 1]
The determinant of matrix A, denoted as |A| or det(A), is calculated as follows:
|A| = (2 * 1) - (5 * 3) = -13
Notation used for determinants
- Determinants are commonly denoted using vertical bars or brackets, such as |A| or [A].
- The vertical bar notation is more commonly used in mathematics.
- The size of the matrix is represented as a subscript within the notation, such as |A| for a 2x2 matrix.
Example:
For a 3x3 matrix B:
B = [1 2 3] [4 5 6] [7 8 9]
The determinant of matrix B can be written as |B| or det(B).
- Introduction to determinants.
- Methods for calculating determinants of 2x2 matrices.
- Direct multiplication method.
- Cross multiplication method.
- Determinants of 3x3 matrices.
- Expansion along rows or columns.
- Cofactor expansion method.
- Properties of determinants.
- Multiplicative property.
- Transpose property.
- Row or column scaling property.
- Applications of determinants.
- Solving systems of linear equations.
- Finding the inverse of a matrix.
- Higher order determinants.
- Cofactor expansion method for larger matrices.
- Laplace expansion method.
Example:
Solve the following system of linear equations using determinants:
``
5x - 2y = -8
By using determinants, we can write the equations in matrix form as:
[2 3] [x] [7]
[5 -2] * [y] = [-8]
``
Then, we can use determinants to find the values of x and y.
Thank you for your attention
- Feel free to ask any questions you may have.
- Next lecture: Solving systems of linear equations using determinants.