Determinants

The determinant is a number that describes a matrix.

The Essentials

To find the determinant of a two by two matrix, take the product of the first and last terms and subtract the product of the other two terms:

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc \]

To find the determinant of a 3 by 3 matrix, choose a row. For each component in that row, cross out the row and column for that component, and multiply the component by the determinant of the numbers not crossed out. These three terms are added together and each are positive or negative depending on where the chosen component is:

\[ \begin{bmatrix} + & - & +\\ - & + & -\\ + & - & + \end{bmatrix} \]

Therefore, if the top row is chosen, the first and third terms will be positive and the second will be negative. Alternatively, a column can be chosen instead of a row. Here is a generic three by three matrix:

\[ \begin{bmatrix} a & b & c\\ d & e & f \\ g & h & i \end{bmatrix} \]

If we choose the top row, the first term in the determinant would be:

\[ a \begin{vmatrix} e & f \\ h & i \end{vmatrix} \]

Where e, f, h, and i are all the values not in the same column or row as a. If the rest of the terms are calculated likewise, the determinant will result in:

\[ a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix} \]
\[ a(ei-fh)-b(di-fg)+c(dh-eg) \]
\[ aei-afh-bdi+bfg+cdh-ceg \]

This method can generalize to higher-dimension matrices. For example, the determinant of a four by four matrix would have four terms, a scalar multiplied by the determinant of a three by three matrix. This matrix would be used to know which terms are negative and which are positive:

\[ \begin{bmatrix} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{bmatrix} \]

Example

Find the determinant of this matrix:

\[ \begin{bmatrix} 1 & 2 & 0 \\ 2 & 4 & 3 \\ 1 & 1 & 1 \end{bmatrix} \]

We will go across the first row:

\[ 1 \begin{vmatrix} 4 & 3 \\ 1 & 1 \end{vmatrix} - 2 \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} + 0 \begin{vmatrix} 2 & 4 \\ 1 & 1 \end{vmatrix} \]

Note that the second term is negative because of its position, and that the third term is zero.

\[ 1(4-3)- 2(2-3)=3 \]

Note that if the zero in the top row of the matrix was in a different row, it might be computationally easier to choose a row other than the first row.

Practice

  1. Find the determinant of this matrix:
    \[ \begin{bmatrix} 1 & 4 & 3 \\ 1 & 2 & 3 \\ 1 & 1 & 1 \end{bmatrix} \]
  2. What value of a will make the determinant of this matrix 8?
    \[ \begin{bmatrix} 1 & 4 & 3 & 0 \\ 1 & a & 3 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 2 \end{bmatrix} \]

Solutions:

  1. 4
  2. 2