17.2: Properties of Determinants

The following are some helpful properties when working with determinants. These properties are often used in proofs and can sometimes be utilized to make faster calculations.

Row Operations

Let \(A\) be an \(n \times n\) matrix and \(c\) be a nonzero scalar. Let \(\left| A \right|\) be a simplified syntax for writing the determinant of \(A\):

1. If a matrix \(B\) is obtained from \(A\) by multiplying a row (column) by \(c\) then \( \left| B \right| = c \left| A \right|\).
2. If a matrix \(B\) is obtained from \(A\) by interchanging two rows (columns) then \(\left| B \right| = − \left| A \right|\).
3. if a matrix \(B\) is obtained from \(A\) by adding a multiple of one row (column) to another row (column), then \(\left| B \right| = \left| A \right| \).

Singular Matrices

Definition: A square matrix \(A\) is said to be singular if \(\left| A \right| = 0\). \(A\) is nonsingular if \(\left| A \right| \neq 0 \)

Now, Let \(A\) be an \(n \times n\) matrix. \(A\) is singular if any of these is true:

1. all the elements of a row (column) are zero.
2. two rows (columns) are equal.
3. two rows (columns) are proportional. i.e. one row (column) is the same as another row (column) multiplied by \(c\).

Question

The following matrix is singular because of certain column or row properties. Give the reason:

\[\begin{split}
\left[
\begin{matrix}
1 & 5 & 5 \\
0 & -2 & -2 \\
3 & 1 & 1
\end{matrix}
\right]
\end{split} \nonumber \]

Question

The following matrix is singular because of certain column or row properties. Give the reason:

\[\begin{split}
\left[
\begin{matrix}
1 & 0 & 4 \\
0 & 1 & 9 \\
0 & 0 & 0
\end{matrix}
\right]
\end{split} \nonumber \]

Determinants and Matrix Operations

Let \(A\) and \(B\) be \(n \times n\) matrices and \(c\) be a nonzero scalar.

Determinant of a scalar multiple: \( \left| cA \right| = c^n \left| A \right| \)

Determinant of a product: \(\left| AB \right| = \left| A \right| \left| B \right | \)

Determinant of a transpose: \(\left| A^t \right| = \left| A \right| \)

Determinant of an inverse: \(\left| A-1 \right| = \frac{1}{\left| A \right|} \) (Assuming \(A-1\) exists)

Question

If \(A\) is a \(3 \times 3\) matrix with \(\left| A \right| = 3\), use the properties of determinants to compute the following determinant:

\[ \left| 2A \right| \nonumber \]

Question

If \(A\) is a \(3 \times 3\) matrix with \( \left| A \right| = 3 \), use the properties of determinants to compute the following determinant:

\[ \left| A^2 \right| \nonumber \]

Question

If \(A\) and \(B\) are \(3 \times 3\) matrices and \( \left| A \right| = -3 \), \( \left| B \right| = 2\), compute the following determinant:

\[ \left| AB \right| \nonumber \]

Question

If \(A\) and \(B\) are \(3 \times 3\) matrices and \( \left| A \right| = -3 \), \( \left| B \right| = 2\), compute the following determinant:

\[ \left| 2 AB^{-1} \right| \nonumber \]

Triangular matrices

Definition: An upper triangular matrix has nonzero elements lie on or above the main diagonal and zero elements below the main diagonal. For example:

\[\begin{split} A =
\left[
\begin{matrix}
2 & -1 & 9 & 4 \\
0 & 3 & 0 & 6 \\
0 & 0 & -5 & 3 \\
0 & 0 & 0 & 1
\end{matrix}
\right]
\end{split} \nonumber \]

The determinant of an upper triangle matrix \(A\) is the product of the diagonal elements of the matrix \(A\).

Also, since the Determinant is the same for a matrix and it’s transpose (i.e. \( \left| A^t \right| = \left| A \right| \), see definition above) the determinant of a lower triangle matrix is also the product of the diagonal elements.

Question

What is the determinant of matrix \(A\)?

Using Properties of determinants:

Here is a great video showing how you can use the properties of determinants:

from IPython.display import YouTubeVideo
YouTubeVideo("aKX5_DucNq8",width=640,height=360, cc_load_policy=True)

Question (A challenging one)

Using the pattern established in the video can you calculate the determinate of the following matrix?

\[\begin{split}
\left[
\begin{matrix}
1 & a & a^2 & a^3 \\
1 & b & b^2 & b^3 \\
1 & c & c^2 & c^3 \\
1 & d & d^2 & d^3
\end{matrix}
\right]
\end{split} \nonumber \]