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Determinants

Definition

  • Every square matrix A has an associated number, called its determinant and denoted by \(\det(A)\) or \(|A|\). It is used in :
    • Solving systems of linear equations
    • Finding the inverse of a matrix
    • Calculus and More

Calculating Determinants

  • Determinant of a 1 x 1 matrix : \(\det(A) = a_{11}\)

  • Determinant of a 2 x 2 matrix :

    • \(A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\)
    • \(det(A) = ad - bc\)

  • Determinant of a 3 x 3 matrix :

    • \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}\)
    • We will the get the determinant by using the first row.
    • \(det(A) = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \\ \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \\ \end{vmatrix}\)
    • \(det(A) = a_{11} (a_{22}a_{33} - a_{23}a_{32}) - a_{12} (a_{21}a_{33} - a_{23}a_{31}) + a_{13} (a_{21}a_{32} - a_{22}a_{31})\)

Example

\(A = \begin{bmatrix} 2 & 4 & 1 \\ 3 & 8 & 7 \\ 5 & 6 & 9 \\ \end{bmatrix}\)

\(\begin{align} \det(A) &= 2 \begin{vmatrix} 8 & 7 \\ 6 & 9 \\ \end{vmatrix} - 4 \begin{vmatrix} 3 & 7 \\ 5 & 9 \\ \end{vmatrix} + 1 \begin{vmatrix} 3 & 8 \\ 5 & 6 \\ \end{vmatrix} \\ &= 2 (8 \times 9 - 7 \times 6) - 4 (3 \times 9 - 7 \times 5) + 1 (3 \times 6 - 8 \times 5) \\ &= 2 (72 - 42) - 4 (27 - 35) + 1 (18 - 40) \\ &= 2 (30) - 4 (-8) + 1 (-22) \\ &= 60 - (-32) + 22 \\ &= 94 \end{align}\)

Determinant of Identity Matrix

  • Determinant of a 1 x 1 identity matrix : \(\det(I) = 1\)
  • Determinant of a 2 x 2 identity matrix : \(\det(I) = (1 \times 1) - (0 \times 0) = 1\)
  • Determinant of a 3 x 3 identity matrix : \(\det(I) = 1 \times 1 \times 1 - 0 \times 0 \times 0 = 1\)

Properties of Determinant

  • Determinant of Product of Matrices : \(\det(AB) = \det(A) \times \det(B)\)
  • Determinant of Inverse of Matrix : \(\det(A^{-1}) = \frac{1}{\det(A)}\)
  • Switching Rows : \(\det(A') = -\det(A)\) ( \(A'\) is the new matrix with switched rows.)
  • Switching Columns : \(\det(A') = -det(A)\) (\(A'\) is the new matrix with switched columns.)

  • Adding multiples of rows: (Same proof for multiple of columns)

    • \(A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\)

    • \(A' = \begin{bmatrix} a + tc & b + td \\ c & d \\ \end{bmatrix}\)

      \(\begin{align} \det(A') &= (a + tc)(d) - (b + td)(c) \\ &= ad + tcd - bc - tcd \\ &= ad - bc \\ &= \det(A) \\ \end{align}\)

  • Scalar multiplication of a row/column of matrix \(A\) with a constant \(t\) : \(det(A') = t \cdot det(A)\)

Minors

  • Minor of a matrix is the determinant of a submatrix obtained by deleting a row and a column from the matrix.
  • Minors are denoted by \(M_{ij}\).

Example

\(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}\)

\(M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{vmatrix}\)

\(M_{11} = a_{22}a_{33} - a_{23}a_{32}\)

Cofactors

  • Cofactor of a matrix is the determinant of a submatrix obtained by deleting a row and a column from the matrix.

  • Cofactors are used to calculate the inverse of a matrix.

  • We can use Minors and Cofactors to calculate the determinant of a matrix.

Formula

\[\sum_{j=1}^n (-1)^{1+i} \cdot a_{1i} \cdot M_{1j} = \sum_{j=1}^n (-1)^{1+i} \cdot a_{1i} \cdot C_{1j}\]

Expansion along any row or column

\[det(A) = \sum_{j=1}^n (-1)^{1+i} \cdot a_{ii} \cdot C_{ij} \text{ (i is fixed)}\]
\[\text{OR}\]
\[det(A) = \sum_{j=1}^n (-1)^{1+i} \cdot a_{ii} \cdot C_{ij} \text{ (j is fixed)}\]

More Properties of Determinants

  • \(\det(A^n) = \det(A)^n\)
  • \(\det(A^{-1}) = \frac{1}{\det(A)}\)
  • \(\det(P^{-1}AP) = \det(A)\)
  • \(\det(AB) = \det(BA)\)
  • \(\det(A^T) = \det(A)\)
  • \(\det(tA_{n\times n}) = t^n \cdot \det(A)\)

Tips for Calculating Determinant

  • The determinant of a matrix with a row or column of zeros is \(0\).
  • The determinant of a matrix in which one row (or column) is a linear combination of other rows (resp. columns) is \(0\).
  • Scalar multiplication of a row by a constant t multiplies the determinant by \(t\).
  • While computing the determinant, you can choose to compute it using expansion along a suitable row or column.