Determinant of a 1×1 matrix

This post explains how to calculate the determinant of a 1×1 matrix, also you will see several examples so that you do not have any doubt how to do it. Although this kind of determinants are very rare, the determinants of 1×1 dimension matrices are very easy to solve, as you will see below.

What is a determinant of a 1×1 matrix?

A 1×1 determinant is a matrix of order 1, that is of a row and a column, represented with a vertical bar at each side of the matrix. For example, the following matrix has a single row and a single column:

\displaystyle A = \begin{pmatrix} 3 \end{pmatrix}

Then, the determinant of matrix A is represented as follows:

\displaystyle \lvert A \rvert = \begin{vmatrix} 3\end{vmatrix}

As you have seen, writing the determinant of a square 1×1 matrix is very simple, since the matrix is formed only by 1 row and 1 column and, therefore, the determinant consists of a single number.

How to find the determinant of a 1×1 matrix?

When the matrix is of dimension 1×1, the determinant of this matrix has only one element. Therefore, the result of the determinant of a 1×1 matrix is that element itself.

Examples of determinants 1×1

Example 1

Calculate the determinant of the following 1×1 matrix:

 \displaystyle A=\begin{pmatrix} 5 \end{pmatrix}

It is a 1×1 matrix, so the determinant of A is equal to the number that contains the matrix:

 |A|= \begin{vmatrix} 5 \end{vmatrix} = \bm{5}

Example 2

Find the determinant of the following 1×1 matrix:

 \displaystyle B=\begin{pmatrix} -2 \end{pmatrix}

It is a square matrix of order 1, so the determinant of B is:

 \displaystyle |B|= \begin{vmatrix} -2 \end{vmatrix} = \bm{-2}

Finding the determinant of a 1×1 matrix is not complicated, but you have to pay attention to the sign of the number.

Do not confuse the determinant of a 1×1 matrix with the absolute value of a number.

The result of a 1×1 determinant is always equal to the value of the matrix, regardless of the sign  \longrightarrow \begin{vmatrix} -4 \end{vmatrix} = -4

Instead, the absolute value transforms the number inside the operator into positive  \longrightarrow \begin{vmatrix}.-4 \end{vmatrix} = +4

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