The Determinant and the Inverse

Determinant

Determinant \(|A|\) is a unique scalar associated with a square matrix \(A\).
 
Determinant of a \(2 \times 2\) Matrix: \[A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\] Can be calculated as: \[|A|=a_{11} a_{22}-a_{12} a_{21}\]

Determinant of a \(3 \times 3\) Matrix

\[\begin{aligned} |A| &=\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right| \\~\\ &=a_{11}\left|\begin{array}{ll} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right|-a_{12}\left|\begin{array}{ll} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array}\right| +a_{13}\left|\begin{array}{ll} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right| \end{aligned}\]

Determinant of a \(n \times n\) Matrix

A minor of the element \(a_{ij}\), denoted by \(|M_{ij}|\) is obtained by deleting the \(i\)th row and \(j\)th column.
 
Cofactor \(C_{ij}\) is defined as: \[|C_{ij}| = (-1)^{i+j} |M_{ij}|\] Then, \[|A| = \sum_{i=1}^n a_{ij} |C_{ij}| = \sum_{j=1}^n a_{ij} |C_{ij}|\]

Find the Determinant

\[A=\left[\begin{array}{ccc} 1 & 5 & 1 \\ 0 & 3 & 9 \\ -1 & 0 & 0 \end{array}\right]\]

Properties of Determinants

• \(|A| = |A^T|\)

• Interchanging rows or columns will alter the sign but not the value

• Multiplication of any one row (or one column) by a scalar \(k\) will change the value of the determinant \(k\)-fold

• The addition (subtraction) of a multiple of any row (or column) to (from) another row (or column) will leave the determinant unaltered

• If one row (or column) is a multiple of another row (or column), the value of the determinant will be zero.

Criteria for Nonsingularity

The following statements are equivalent:

• \(|A| \neq 0\)

• Rows (or equivalently columns) of \(A\) are independent

• \(A\) is nonsingular

• \(A^{-1}\) exists

• A unique solution to \(A x=b\) (\(x^*=A^{-1}b\)) exists

Matrix Inversion

Adjoint of a nonsingular \(n \times n\) matrix: \[adj A = C^T = \left[\begin{array}{llll} |C_{11}| & |C_{21}| & \cdots & |C_{n1}| \\ |C_{12}| & |C_{22}| & \cdots & |C_{n2}| \\ \vdots &\vdots & \cdots & \vdots \\ |C_{1n}| & |C_{2n}| & \cdots & |C_{nn}| \\ \end{array}\right]\]

The inverse of \(A\) is defined as: \[A^{-1} = \frac{1}{|A|} \, adj A\]

Find the Inverse

\[A=\left[\begin{array}{ccc} 3 & 2 \\ 1 & 0 \\ \end{array}\right]\]

Find the Inverse

\[A=\left[\begin{array}{ccc} 1 & 5 & 1 \\ 0 & 3 & 9 \\ -1 & 0 & 0 \end{array}\right]\]

Homework Questions

• Exercise 5.2: 1 (c), (e), (f), 2, 3, 6

• Exercise 5.3: 1, 4, 5, 8

• Exercise 5.4: 2, 3, 4, 7

Textbook reference: 5.2-5.4