Matrices and Determinants Cheatsheet

Cheat Codes & Shortcuts

• Matrix: Rectangular array of numbers, e.g., \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
• Determinant (2x2): For \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), \( \det(A) = ad - bc \).
• Determinant (3x3): Use cofactor expansion or rule of Sarrus.
• Matrix Addition: \( A + B \), add corresponding elements, same dimensions.
• Matrix Multiplication: \( AB \), row-by-column, if dimensions compatible.
• Inverse (2x2): For \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), \( A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
• Adjoint: Transpose of cofactor matrix.
• Cramer’s Rule: Solve \( Ax = b \), \( x_i = \frac{\det(A_i)}{\det(A)} \).
• Rank: Number of linearly independent rows/columns.
• Eigenvalues: Solve \( \det(A - \lambda I) = 0 \).

Quick Reference Table

Type	Form	Solution/Method
Determinant	\( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)	\( \det(A) = 1 \cdot 4 - 2 \cdot 3 = -2 \)
Inverse	\( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)	\( A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \)
Cramer’s Rule	\( \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 4 \end{bmatrix} \)	Compute \( \det(A) \), \( \det(A_1) \), \( \det(A_2) \)
Eigenvalues	\( \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \)	Solve \( \det \begin{bmatrix} 3 - \lambda & 1 \\ 1 & 3 - \lambda \end{bmatrix} = 0 \)
Rank	\( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \)	Row reduce to find independent rows
Matrix Product	\( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)	Row-by-column multiplication

Advice

Matrices and Determinants Quick Tips

• Matrix Addition: Only for same dimensions, add element-wise.
• Determinant: For 2x2, use \( ad - bc \); for 3x3, use Sarrus or expansion.
• Inverse: Compute using adjoint and determinant, check \( AA^{-1} = I \).
• Eigenvalues: Solve characteristic equation \( \det(A - \lambda I) = 0 \).
• Rank: Use row reduction to find number of independent rows.

Matrices and Determinants Speed Quiz

Test your speed with 5 matrices and determinants questions! You have 30 seconds per question.