We do not discuss basic knowledge of matrices in great detail here since everything has been discussed in MATH 115A Chapter 2. This note merely serves as a quick review.
Still, readers should pay attention to determinants of the matrices, and $p $th order-minor of matrices, since they are new contents in this class.
A matrix is a rectangular array of numbers. A matrix $\b{ A } $ with $m $ rows and $n $ columns is called an $m \times n $ matrix, and we write
$$\b{ A } = {\Mqr{{\vcr{ a_{11}}{ a_{21}}{ … }{ a_{m1}}}}{{\vcr{ a_{12}}{ a_{22}}{ … }{ a_{2n}}}}{{\vcr{ … }{ \vdots }{ \ddots }{ \vdots }}}{{\vcr{ a_{1n}}{ a_{m2}}{ … }{ a_{mn}}}}} $$
For convenience, we denote the $i $th column of $\b{ A } $ as $\b{ a }_i $ in this note.
The maximal number of linearly independent columns of $\b{ A } $ is called the rank of the matrix $\b{ A } $, denoted $\rank{ \b{ A }} $. Note that $\rank{ \b{ A }} $ is the dimension of $\spa{ \set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ n }}} $, where $\b{ a_i } $ is $i $th column of the matrix $\b{ A } $.
The rank of a matrix $\b{ A } $ is invariant under the following operations:
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Multiplication of the columns or rows of $\b{ A } $ by nonzero scalars.
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Interchange of the columns or rows.
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Addition to a given column (row) a linear combination of other columns (rows).
$$\begin{align*} & \det \left [ {\vce{ \mid }{ \b{ a }_1 }{ \mid }}, …, {\vce{ \mid }{ \b{ a }_{k-1}}{ \mid }}, {\vce{ \mid }{ \pare{p \b{ a }_{k}^{(1)} + q \b{ a }_k^{(2)}}}{ \mid }}, {\vce{ \mid }{ \b{ a }_{k+1}}{ \mid }}, …, {\vce{ \mid }{ \b{ a }_n }{ \mid }} \right ] \br &= \det \left [ {\vce{ \mid }{ \b{ a }_1 }{ \mid }}, …, {\vce{ \mid }{ p \b{ a }_{k}^{(1)}}{ \mid }}, …, {\vce{ \mid }{ a_n }{ \mid }} \right ] + \det \left [ {\vce{ \mid }{ \b{ a }_1 }{ \mid }}, …, {\vce{ \mid }{q \b{ a }_k^{(2)}}{ \mid }}, …, {\vce{ \mid }{ \b{ a }_n }{ \mid }} \right ] \end{align*}$$
- If $\b{ a }_{k} = \b{ a }_{k+1} $, then $\det \b{ A } = 0 $.
- If $\b{ a }_{k} = 0 $, then $\det \b{ A } = 0 $.
- $\det \b{ I } = 1 $.
- The determinant changes its sign if we interchange columns.
- The determinant does not change if one column is added with another column’s multiple.
- $p \det \left [ {\vce{ \mid }{ \b{ a }_1 }{ \mid }}, …, {\vce{ \mid }{ \b{ a }_j}{ \mid }},…, {\vce{ \mid }{ \b{ a }_n }{ \mid }} \right ] = \det \left [ {\vce{ \mid }{ \b{ a }_1 }{ \mid }}, …, {\vce{ \mid }{ p\b{ a }_j}{ \mid }},…, {\vce{ \mid }{ \b{ a }_n }{ \mid }} \right ]$
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A $p $th-order minor of an $m \times n $ matrix $\b{ A } $, with $p < \min{ m, n } $, is the determinant of a $p \times p $ matrix obtained from $\b{ A } $ by deleting $m - p $ rows and $n - p $ columns.
If an $m \times n (m \geq n)$ matrix $A $ has a nonzero $n $th-order minor, then the columns of $\b{ A } $ are linearly independent, that is $\rank{ \b{ A }} = n $.
An invertible matrix is a square matrix whose determinant is nonzero.