Notes on Determinants (Lecture 1, MATH123/132)

2x2 determinants

• A determinant is a number associated with a square matrix that encodes certain directional and scaling properties of the linear transformation represented by the matrix.
• 2×2 determinant formula:
  • For a matrix A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, the determinant is
  • $\det(A) = ad - bc.$
• Example:
  • If A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, then
  • $\det(A) = 1\cdot 4 - 2\cdot 3 = 4 - 6 = -2.$
• Use in solving problems: the determinant provides a quick test for invertibility (nonzero det => invertible) and encodes area scaling for the linear map in R^2.

2x2 determinants and eigenvalues (context for the lecture)

• For a 2×2 matrix A, the eigenvalues λ satisfy the characteristic equation det(A - λI) = 0.
• For a 2×2 matrix, det(A - λI) expands to a quadratic:
  • If tr(A) denotes the trace and det(A) the determinant of A, then the characteristic polynomial is
  • $\lambda^2 - (\operatorname{tr} A)\,\lambda + \det(A) = 0.$
• Solving this quadratic using the quadratic formula gives the eigenvalues:
  • $\lambda = \frac{\operatorname{tr} A \pm \sqrt{(\operatorname{tr} A)^2 - 4\det(A)}}{2}.$
• Example tied to the lecture content (illustrating the quadratic formula usage): if the characteristic polynomial is $\lambda^2 + 3\lambda + 2 = 0,$ then the roots are
  • $\lambda_{1,2} = \frac{-3 \pm \sqrt{9 - 8}}{2} = -1, -2.$
• Note: the ab (abc) form i.e. the quadratic formula is what is used to solve the characteristic equation in this context.

3x3 determinants

• General rule: for a 3×3 matrix A = (a_{ij}), you can compute det(A) by cofactor expansion (Laplace expansion) along any row or column.
• Cofactor expansion along the first row:
• $\det(A) = a<em>{11}\det(M</em>{11}) - a<em>{12}\det(M</em>{12}) + a<em>{13}\det(M</em>{13}),$
  where M_{ij} is the (i,j)-minor obtained by deleting row i and column j.
• Example from the lecture (worked 3×3 determinant):
  • Consider A = \begin{bmatrix} 1 & 2 & 0 \ 3 & -1 & 2 \ 2 & 0 & 1 \end{bmatrix}.
  • Compute minors:
  • M{11} = \begin{bmatrix}-1 & 2 \ 0 & 1\end{bmatrix}, \det(M{11}) = (-1)(1) - (2)(0) = -1.
  • M{12} = \begin{bmatrix}3 & 2 \ 2 & 1\end{bmatrix}, \det(M{12}) = 3\cdot 1 - 2\cdot 2 = -1.
  • M{13} = \begin{bmatrix}3 & -1 \ 2 & 0\end{bmatrix}, \det(M{13}) = 3\cdot 0 - (-1)\cdot 2 = 2.
  • Then
  • $\det(A) = 1\cdot(-1) - 2\cdot(-1) + 0\cdot(2) = -1 + 2 + 0 = 1.$
• It is common to choose the expansion along the first row, but any row or column works.
• Interpretation: the signs for cofactors follow the chessboard pattern (+, -, +; -, +, -; +, -, +), i.e., the sign of C_{ij} is $(-1)^{i+j}$.

Reduction algorithm and cofactor expansion (kofaktor-utvikling)

• Main idea: compute det(A) by reducing to a form where the determinant is easy to read off.
• Two related approaches discussed in the lecture:
  • Reduction to triangular form via row operations (Gaussian elimination) and then take the product of the diagonal entries; track sign changes and scaling changes caused by row operations.
  • Cofactor (Laplace) expansion along a row or a column as a general method (kofaktor-utvikling).
• Sample worked path from the lecture:
  • Consider A = \begin{bmatrix} 1 & 2 & 0 \ 3 & -1 & 2 \ 2 & 0 & 1 \end{bmatrix}.
  • Cofactor expansion along the first row (row 1):
  • $\det(A) = a<em>{11}\det(M</em>{11}) - a<em>{12}\det(M</em>{12}) + a<em>{13}\det(M</em>{13}).$
  • Substituting the values from the previous section yields det(A) = 1. The explicit calculation as shown above confirms det(A) = 1.
• Fortegnene-regelen (sign rule) and the sjakkbrett-regelen (chessboard pattern) are essential for correct cofactor signs.

Determinants of triangular matrices

• If a matrix is upper triangular (all entries below the main diagonal are zero) or lower triangular (all entries above the main diagonal are zero), then the determinant is the product of the diagonal entries:
• $\det(A) = \prod<em>{i=1}^n a</em>{ii}.$
• Rounding out the same result for triangular matrices in the 3×3 case: example with a triangular matrix shows the determinant equals the product of diagonal elements.
• In the lecture, an example shows computation by triangular form and confirms the product-of-diagonal rule in the triangular case.

General properties and key rules

• Determinants multiply across matrix products:
• $\det(AB) = \det(A)\,\det(B).$
• Det(A+B) is not generally equal to det(A) + det(B) (linearity does not extend to det over matrix addition).
• Invertibility criterion:
  • A square matrix A is invertible iff $\det(A) \neq 0.$
  • Equivalently, det(A) = 0 iff A is not invertible.
• Transpose invariance:
  • $\det(A^{\top}) = \det(A).$
• Row and column operations and their effects:
  • Swapping two rows (or two columns) changes the sign of the determinant.
  • Adding a multiple of one row to another does not change the determinant.
  • Scaling a row by a nonzero scalar multiplies the determinant by that scalar (same for columns).
• More broadly, the determinant is a measure of volume scaling of the associated linear transformation and encodes invertibility and orientation effects.
• The determinant also satisfies the eigenvalue relation through the characteristic polynomial det(A - \lambda I) = 0, which in the 2×2 case reduces to a quadratic equation as described above.

Summary and connections to broader topics

• Determinants can be computed only for square matrices.
• The reduction algorithm (via cofactor expansion or via row-reduction to triangular form) is a general method for computing determinants.
• For small sizes, 2×2 and 3×3, there are simple, explicit formulas:
  • 2×2: $\det\begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc.$
  • 3×3: use the cofactor expansion along a row or column or a standard 3×3 rule (as illustrated).
• The determinant of a triangular matrix equals the product of its diagonal elements.
• A is invertible exactly when $\det(A) \neq 0.$ The determinant also obeys the rules $\det(AB) = \det(A)\det(B)$ and $\det(A^{\top}) = \det(A).$
• The sign pattern for cofactors is given by the checkerboard rule $(-1)^{i+j}$, which is often remembered as the chessboard rule.
• In eigenvalue problems, the characteristic polynomial det(A - \lambda I) = 0 leads to a quadratic for 2×2 matrices, solved by the quadratic formula, yielding eigenvalues such as $\lambda = \frac{\operatorname{tr}(A) \pm \sqrt{(\operatorname{tr}(A))^2 - 4\det(A)}}{2}.$
• Practical interpretation: determinants provide a compact summary of properties like invertibility, volume scaling, and orientation changes under linear transformations.