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Understanding Determinants

Understanding Determinants: A Complete Guide

The determinant of a matrix is a special number that provides important geometric and algebraic information. It helps us determine if a matrix is invertible, and it describes transformations such as **scaling, flipping, and dimension collapse**.

Key Interpretations of the Determinant:
  • Area Factor: In 2D, the determinant represents the scaling factor of areas under transformation.
  • Flipping Shapes: A negative determinant means the shape is flipped (reflection).
  • Collapsing Dimensions: If the determinant is zero, the transformation collapses to a lower dimension.

Determinant of a 2×2 Matrix

For a 2×2 matrix:

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

The determinant is given by:

\[ \det(A) = ad - bc \]

Example:

Find the determinant of:

\[ A = \begin{bmatrix} 3 & 2 \\ 5 & 4 \end{bmatrix} \]

Using the formula:

\[ \det(A) = (3 \times 4) - (2 \times 5) = 12 - 10 = 2 \]

Determinant of a 3×3 Matrix

For a 3×3 matrix:

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

The determinant is found using expansion along the first row:

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

Example:

Find the determinant of:

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix} \]

Compute the minors:

\[ \det(A) = 1 \begin{vmatrix} 5 & 6 \\ 8 & 10 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6 \\ 7 & 10 \end{vmatrix} + 3 \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} \]

Calculate each minor:

\[ \begin{vmatrix} 5 & 6 \\ 8 & 10 \end{vmatrix} = (5 \times 10) - (6 \times 8) = 50 - 48 = 2 \] \[ \begin{vmatrix} 4 & 6 \\ 7 & 10 \end{vmatrix} = (4 \times 10) - (6 \times 7) = 40 - 42 = -2 \] \[ \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = (4 \times 8) - (5 \times 7) = 32 - 35 = -3 \]

Final calculation:

\[ \det(A) = (1 \times 2) - (2 \times -2) + (3 \times -3) = 2 + 4 - 9 = -3 \]

Practice Problems

Problem 1: Find the determinant of

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

\(\det(A) = (2 \times 4) - (3 \times 1) = 8 - 3 = 5\)

Problem 2: Find the determinant of

\[ A = \begin{bmatrix} 3 & 1 & 4 \\ 2 & 5 & 6 \\ 1 & 2 & 3 \end{bmatrix} \]

\(\det(A) = 3(5 \times 3 - 6 \times 2) - 1(2 \times 3 - 6 \times 1) + 4(2 \times 2 - 5 \times 1)\)

\(= 3(15 - 12) - 1(6 - 6) + 4(4 - 5)\)

\(= 3(3) - 1(0) + 4(-1)\)

\(= 9 + 0 - 4 = 5\)

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