1. DETERMINANTS AND MATRICES

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1. DETERMINANTS AND MATRICES

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1.1 Algebra of Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. The basic operations that can be performed on matrices are addition, subtraction, and multiplication.

Matrix Addition:

Two matrices can be added if they have the same dimensions. Simply add the corresponding elements.

For example, if:

$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$$

Then:

$$A + B = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$

Matrix Subtraction:

Matrix subtraction follows the same rule as matrix addition: subtract corresponding elements.

For example:

$$A - B = \begin{pmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{pmatrix} = \begin{pmatrix} -4 & -4 \\ -4 & -4 \end{pmatrix}$$

Matrix Multiplication:

Matrix multiplication is more complex. You multiply each row of the first matrix by each column of the second matrix. The number of columns of the first matrix must equal the number of rows of the second matrix.

For example, if:

$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$$

Then the product $A \times B$ is:

$$A \times B = \begin{pmatrix} (1*5 + 2*7) & (1*6 + 2*8) \\ (3*5 + 4*7) & (3*6 + 4*8) \end{pmatrix}$$
$$= \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$$

Transpose of a Matrix:

The transpose of a matrix is formed by swapping its rows and columns. Denoted by $A^T$.

For example:

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

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1.2 Elementary Properties of Determinants (up to 3rd Order)

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties such as the matrix’s invertibility and solutions to linear systems.

Determinants of 2x2 Matrix:

For a 2x2 matrix:

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

The determinant is:

$$\text{det}(A) = ad - bc$$

Determinants of 3x3 Matrix:

For a 3x3 matrix:

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

The determinant is calculated using:

$$\text{det}(A) = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}$$

Expanding the 2x2 determinants:

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

Properties of Determinants:

1. If two rows or columns are interchanged, the sign of the determinant changes.
2. If a row or column is multiplied by a scalar, the determinant is multiplied by that scalar.
3. If two rows or columns are identical, the determinant is 0.

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1.3 Consistency of Equations, Cramer's Rule

The consistency of a system of linear equations refers to whether the system has a solution. A system is consistent if it has at least one solution (either one or infinitely many). If the determinant of the coefficient matrix is non-zero, the system has a unique solution.

Cramer's Rule provides a way to solve a system of linear equations using determinants.

For a system of equations:

$$Ax = b$$

Where $A$ is the coefficient matrix, $x$ is the column matrix of variables, and $b$ is the constants matrix.

For a 2x2 system:

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \quad b = \begin{pmatrix} e \\ f \end{pmatrix}$$

The solution is:

$$x_1 = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\text{det}(A)}, \quad x_2 = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\text{det}(A)}$$

For a 3x3 system, the solution for each variable involves replacing the corresponding column of the coefficient matrix with the constants matrix and computing the determinant.

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1.4 Inverse of a Matrix

The inverse of a matrix $A$ is denoted by $A^{-1}$, and it satisfies the equation:

$$A \times A^{-1} = I$$

Where $I$ is the identity matrix.

Inverse of a 2x2 Matrix:

For a matrix:

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

The inverse is given by:

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

Where $\text{det}(A)=ad-bc$

Inverse of a 3x3 Matrix:

The inverse of a 3x3 matrix is more complicated and involves finding the matrix of cofactors, adjugates, and then dividing by the determinant.

For a matrix $A$, the inverse is:

$$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$$

Where $\text{adj}(A)\; is\; the\; adjugate\; matrix,\; which\; is\; the\; transpose\; of\; the\; cofactor\; matrix.$

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1.5 Inverse Matrix Method to Solve a System of Linear Equations in 3 Variables

For a system of linear equations:

$$Ax = b$$

Where $A$ is the coefficient matrix, $x$ is the column matrix of variables, and $b$ is the constants matrix.

The solution can be found by:

$$x = A^{-1} b$$

Thus, you first calculate the inverse of the matrix $A$ and then multiply it by the constants matrix $b$.

Example: Consider the system:

$$x + 2y + z = 9$$
$$2x + y + 3z = 8$$
$$3x + 4y + z = 7$$

The coefficient matrix is:

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

And the constants matrix is:

$$b = \begin{pmatrix} 9 \\ 8 \\ 7 \end{pmatrix}$$

1. Compute the inverse of matrix $A$.
2. Multiply the inverse of $A$ by $b$ to get the solution vector $x$.

The final result will give you the values for $x$, $y$, and $z$.

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Summary of Formulas:

1. Matrix Addition/Subtraction:

   • $A + B = \begin{pmatrix} a+b & c+d \\ e+f & g+h \end{pmatrix}$
     

2. Matrix Multiplication:

   • $A \times B = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \times \begin{pmatrix} e & f \\ g & h \end{pmatrix}$
     

3. Determinant of 2x2 Matrix:

   • $\text{det}(A) = ad - bc$
     

4. Cramer's Rule (2x2 system):

   • $x_1 = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\text{det}(A)}$ $x_2 = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\text{det}(A)}$

5. Inverse of 2x2 Matrix:

• $A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
  

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