In this section, we will delve into the fundamental concepts of matrices and determinants, which are crucial for understanding and manipulating three-dimensional graphics. Matrices are essential tools in linear algebra, used to represent and solve systems of linear equations, perform linear transformations, and more. Determinants, on the other hand, provide valuable information about the properties of matrices, such as invertibility and volume scaling.

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are denoted by uppercase letters (e.g., \(A\), \(B\), \(C\)) and their elements are typically denoted by lowercase letters with two subscripts (e.g., \(a_{ij}\) represents the element in the \(i\)-th row and \(j\)-th column of matrix \(A\)).

Types of Matrices

• Square Matrix: A matrix with the same number of rows and columns (e.g., \(3 \times 3\)).
• Row Matrix: A matrix with a single row.
• Column Matrix: A matrix with a single column.
• Zero Matrix: A matrix in which all elements are zero.
• Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere.

Matrix Notation

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23}
a_{31} & a_{32} & a_{33} \end{pmatrix} \]

Addition and Subtraction

Matrices of the same dimensions can be added or subtracted element-wise.

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

Scalar Multiplication

Each element of the matrix is multiplied by a scalar (a constant).

\[ cA = c \begin{pmatrix} a_{11} & a_{12}
a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} ca_{11} & ca_{12}
ca_{21} & ca_{22} \end{pmatrix} \]

Matrix Multiplication

The product of two matrices \(A\) and \(B\) is defined if the number of columns in \(A\) is equal to the number of rows in \(B\).

\[ C = AB \] \[ c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \]

The determinant is a scalar value that can be computed from a square matrix. It provides important properties about the matrix, such as whether it is invertible.

Determinant of a 2x2 Matrix

\[ \text{det}(A) = \begin{vmatrix} a & b
c & d \end{vmatrix} = ad - bc \]

Determinant of a 3x3 Matrix

\[ \text{det}(A) = \begin{vmatrix} a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23}
a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

• Multiplicative Property: \(\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)\)
• Invertibility: A matrix \(A\) is invertible if and only if \(\text{det}(A) \neq 0\).
• Transpose: \(\text{det}(A^T) = \text{det}(A)\)

Given matrices \(A\) and \(B\):

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

Calculate \(A + B\):

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

Given matrix \(C\):

\[ C = \begin{pmatrix} 1 & 2 & 3
4 & 5 & 6
7 & 8 & 9 \end{pmatrix} \]

Calculate \(\text{det}(C)\):

\[ \text{det}(C) = 1(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7) \] \[ = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) \] \[ = 1(-3) - 2(-6) + 3(-3) \] \[ = -3 + 12 - 9 \] \[ = 0 \]

Given matrices \(A\) and \(B\):

\[ A = \begin{pmatrix} 2 & 0
1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 4
2 & 5 \end{pmatrix} \]

Calculate \(AB\).

\[ AB = \begin{pmatrix} 2 & 0
1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 4
2 & 5 \end{pmatrix} = \begin{pmatrix} 2 \cdot 1 + 0 \cdot 2 & 2 \cdot 4 + 0 \cdot 5
1 \cdot 1 + 3 \cdot 2 & 1 \cdot 4 + 3 \cdot 5 \end{pmatrix} = \begin{pmatrix} 2 & 8
7 & 19 \end{pmatrix} \]

Given matrix \(D\):

\[ D = \begin{pmatrix} 3 & 8
4 & 6 \end{pmatrix} \]

Calculate \(\text{det}(D)\).

\[ \text{det}(D) = 3 \cdot 6 - 8 \cdot 4 = 18 - 32 = -14 \]

In this section, we covered the basics of matrices and determinants, including their definitions, types, operations, and properties. We also provided practical examples and exercises to reinforce the concepts. Understanding matrices and determinants is essential for further topics in linear algebra and 3D graphics, as they form the foundation for more advanced operations and transformations.