Notes#

  • Some properties and behaviors on determinants:

    • Interchange of two rows multiplies the value of the determinant by -1.
    • Addition of a multiple of a row to another row does not alter the value of the determinant.
    • Multiplication of a row by a nonzero constant \(c\) multiplies the value of the determinant by \(c\). (This holds also when \(c=0\), but no longer gives an elementary row operation.)
    • Transposition leaves the value of the determinant unaltered.
    • A zero (redundant) row or column makes the value of the determinant zero.

  • A determinant of the second order is defined by:

\[ D = \text{det }\mathbf{A} = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{12} - a_{12}a_{21} \]
  • A determinant of the third order can be defined by:
\[ D = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{21} \begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} + a_{31} \begin{vmatrix} a_{12} & a_{13} \\ a_{22} & a_{23} \end{vmatrix} \]
  • General form to calculate determinants:
\[ D = \sum_{j=1}^{n} (-1)^{j+k} a_{jk}M_{jk} \qquad (k = 1, 2, \dots n) \]
  • The “sub-determinants” inside of a determinant is called a minor (denoted as \(M_{jk}\)).
  • If a matrix has a rank \(r\) greater than or equal to 1, then there exists at least one \(r \times r\) square submatrix whose determinant is non-zero.
  • If the rank of a matrix is \(r\), then any square matrix that is larger than \(r \times r\) will have a zero determinant.
  • An \(n \times n\) square matrix \(A\) has a rank \(n\) if and only if \(\text{det} A \neq 0\).
  • If the determinant of a matrix \(A\) is 0, then
    • \(A\) does not have an inverse
    • There are redundant rows/columns
  • The difference between rank and full rank:
    • The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
    • A matrix is said to be full rank when its rank is as large as possible for its shape.

Exercises#

Problem 1#

Evaluate the following determinant:

\[\begin{vmatrix} 4 & 1 & 8 \\ 0 & 2 & 3 \\ 0 & 0 & 5 \end{vmatrix}\]
\[ \begin{vmatrix} 4 & 1 & 8 \\ 0 & 2 & 3 \\ 0 & 0 & 5 \end{vmatrix} \]\[\begin{aligned} &= 4\begin{vmatrix} 2 & 3 \\ 0 & 5 \end{vmatrix} - 1\begin{vmatrix} 0 & 3 \\ 0 & 5 \end{vmatrix} + 8\begin{vmatrix} 0 & 2 \\ 0 & 0 \end{vmatrix} \\ &= 4(2 \cdot 5 - 0 \cdot 3) - (0 \cdot 5 - 0 \cdot 3) + 8(0 \cdot 0 - 0 \cdot 2) \\ &= 4(10) -0 + 0 \\ &= 40 \end{aligned}\]

Problem 2#

Evaluate the determinant of the following matrix:

\[\begin{bmatrix} 1 & 5 & 2 & 2 \\ 1 & 3 & 2 & 6 \\ 4 & 0 & 8 & 48 \end{bmatrix}\]

Skipping