Definition 1.5.1.
The rank of a matrix \(A\in M_{m\times n}(\mathbb{K})\) is the number of nonzero rows in the row reduced echelon form of \(A\text{.}\)
We denote the rank of \(A\) by \({\rm rank}(A)\text{.}\)
Section 1.5 Some invariants attached to a matrix
In this section we denote by \(\mathbb{K}\) either the set of real numbers, \(\mathbb{R}\) or the set of complex numbers, \(\mathbb{C}\text{.}\)
Remark 1.5.2.
It follows from Definition 1.5.1 that for a matrix \(A\in M_{m\times n}(\mathbb{K})\text{,}\) the rank of \(A\) is equal to the number of pivots in the row reduced echelon form of \(A\text{,}\) which in turn can be at most \(\min\{m,n\}\text{.}\)
Definition 1.5.3.
The trace of a square matrix is the sum of all its diagonal entries. The trace of a square matrix \(A\in M_{n}(\mathbb{K})\) is denoted by \({\rm tr}(A)\text{.}\) If \(A\in M_{n}(\mathbb{K})\) is given by
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}
\end{equation*}
then the trace of \(A\text{,}\)
\begin{equation*}
{\rm tr}(A)=a_{11}+a_{22}+\cdots+a_{nn}.
\end{equation*}
We now define the “determinant” of a square matrix recursively. We will not give a general definition.
Convention 1.5.4.
Given an \(n\times n\) matrix over \(\K\text{,}\)
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}
\end{equation*}
its determinant is denoted by
\begin{equation*}
\det A=\begin{vmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{vmatrix}.
\end{equation*}
Note 1.5.5.
Apart from various explantions given in lectures one may see this YouTube video 1 with excellent animations to get started with the concept of determinants.
Definition 1.5.6.
The determinant of a matrix \(A=\begin{pmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22}\end{pmatrix}\in M_2(\K)\) is denote by
\begin{equation*}
\det A=\begin{vmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22}\end{vmatrix}
\end{equation*}
and it is equal to \(a_{11}a_{22}-a_{12}a_{21}\in\K\text{,}\) i.e.,
\begin{equation*}
\det A=\begin{vmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22}\end{vmatrix}= a_{11}a_{22}-a_{12}a_{21}\in\K.
\end{equation*}
Definition 1.5.7.
Consider \(A\in M_3(\K)\) as follows.
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp a_{13}\\a_{21}\amp a_{22}\amp a_{23}\\a_{31}\amp a_{32}\amp a_{33}\end{pmatrix}
\end{equation*}
The determinant of \(A\) is
\begin{align*}
\det A\amp=a_{11}\,\begin{vmatrix}a_{22}\amp a_{23}\\a_{32}\amp a_{33}\end{vmatrix}-a_{12}\,\begin{vmatrix}a_{21}\amp a_{23}\\a_{31}\amp a_{33}\end{vmatrix} + a_{13}\, \begin{vmatrix}a_{21}\amp a_{22}\\a_{31}\amp a_{32}\end{vmatrix}\in\K.
\end{align*}
Note 1.5.8.
In the formula of the determinant of \(A\in M_3(\K)\) observe the following.
- Change of signs in consecutive terms.
- First entry is obtained by multiplying \(a_{11}\) by the determinant of \(2\times 2\) matrix obtained by deleting the first row and the first column of \(A\text{.}\) The second entry is obtained by multiplying \(-a_{12}\) by the determinant of \(2\times 2\) matrix obtained by deleting the first row and the second column of \(A\text{.}\) The third entry is obtained by multiplying \(a_{13}\) by the determinant of \(2\times 2\) matrix obtained by deleting the first row and the third column of \(A\text{.}\)
For the rest of this section we use a short notation to write a square matrix, viz., we write \(A=(a_{ij})\in M_n(\K)\) to denote the following matrix.
\begin{equation}
A=(a_{ij})=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}\tag{1.5.1}
\end{equation}
Note 1.5.9.
We stress that the determinant of a square matrix is an element of \(\K\text{.}\)
Fact 1.5.10.
Properties of the determinant. Assume that \(A=(a_{ij})\in M_n(\K)\text{.}\)
- Row linearity. Let \(A_i=(a_{i1}\,a_{i2}\,\cdots\,a_{in})\) be the \(i\)-th row of \(A\text{.}\) For \(\beta\in \K\) we let \(\beta A_i=(\beta a_{i1}\,\beta a_{i2}\,\cdots\,\beta a_{in})\text{.}\) For any \(B\in M_{1\times n}(\K)\) and any \(\beta,\gamma\in \K\) and any \(i\in\{1,2,\ldots,n\}\) we get the following.\begin{equation*} \det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\\beta A_i+\gamma B\\A_{i+1}\\\vdots\\A_n\end{pmatrix}=\beta\,\det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\A_i\\A_{i+1}\\\vdots\\A_n\end{pmatrix}+\gamma\,\det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\B\\A_{i+1}\\\vdots\\A_n\end{pmatrix} \end{equation*}
- For \(A\in M_n(\K)\) and any \(\beta\in \K\text{,}\) \(\det(\beta A)=\beta^n\cdot\det A\text{.}\)
- Row rearrangement. Let \(A^\prime\) be the matrix obtained by interchanging the \(i\)-th row of \(A\) with the \(j\)-th row of \(A\text{.}\) Then\begin{equation*} \det A^\prime=-\det A. \end{equation*}
- Alternating. If any two rows of \(A\in M_n(\K)\) are the same then \(\det A=0\text{.}\)
- Transpose. For any \(A\in M_n(\K)\text{,}\)\begin{equation*} \det A^t=\det A. \end{equation*}
- Triangular matrices. If \(A=(a_{ij})\in M_n(\K)\) is an upper triangular (resp., lower triangular) matrix, i.e., \(a_{ij}=0\) for \(i>j\) (resp., \(a_{ij}=0\) for \(i<j\)) then\begin{equation*} \det A=a_{11}a_{22}\cdots a_{nn}. \end{equation*}
- Block form. Let \(r\in\{1,2,\ldots,n-1\}\text{.}\) Let \(B\in M_r(\K)\text{,}\) \(C\in M_{r\times n-r}(\K)\text{,}\) \(D\in M_{n-r}(\K)\text{,}\) and \(\mathbf{0}\in M_{n-r\times r}(\K)\) be the zero matrix. The determinant of\begin{equation*} A=\begin{pmatrix}B\amp C\\\mathbf{0}\amp D\end{pmatrix} \end{equation*}is given by\begin{equation*} \det A=\det B\cdot\det D. \end{equation*}Similar result is true for lower triangular block matrices.
- Multiplicative property. Let \(A, B\in M_n(\K)\text{.}\) We have\begin{equation*} \det AB=\det A\cdot\det B. \end{equation*}
- Invertibility. A matrix \(A\in M_n(\K)\) is invertible if and only if \(\det A\neq 0\) if and only if \({\rm rank}(A)=n\text{.}\)
