## Determinants as Area and Volume

The Area of a Parallelogram Let $v=(v_1,v_2)$ and $w=(w_1,w_2)$ be two linearly independent vectors in $\mathbb{R}^2$. Then they span a parallelogram as shown in Figure 1. The area $A$ of the parallelogram is \begin{align*} A&=||v||||w||\sin\theta\\ &=||v\times w||\\ &=\left|\begin{array}{cc} v_1 & … Continue reading

## Inverse of a Matrix

Let $A$ be an $n\times n$ matrix with $\det A\ne 0$. (A square matrix whose determinant is not equal to $0$ is called non-singular.) Let $X=(x_{ij})$ be an unknown $n\times n$ matrix such that $AX=I$. Then $$x_{1j}A^1+\cdots+x_{nj}A^n=E^j.$$ This is a … Continue reading

## Cramer’s Rule

Consider a system of $n$ linear equations in $n$ unknowns $$x_1A^1+\cdots+x_nA^n=B,$$ where $x_1,\cdots,x_n$ are variables and $A^1,\cdots,A^n,B$ are column vectors of dimension $n$. Suppose that $\det(A^1,\cdots,A^n)\ne 0$. Recall that this condition ensures that the linear system has a unique solution … Continue reading

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## The Rank of a Matrix 2

A test for the linear dependence of vectors may be given in terms of determinant. Theorem. Let $A^1,\cdots,A^n$ be column vectors of dimension $n$. They are linearly dependent if and only if $$\det(A^1,\cdots,A^n)=0.$$ Corollary. If a system of $n$ linear … Continue reading

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## Determinants II: Determinants of Order $n$

A determinant of order $n$ can be calculated by expanding it in terms of determinants of order $n-1$. Let $A=(a_{ij})$ be an $n\times n$ matrix and let us denote by $A_{ij}$ the $(n-1)\times (n-1)$ matrix obtained by deleting the $i$-th … Continue reading

## Determinants I: Determinants of Order 2

Let $A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}$. Then we define the determinant $\det A$ by $$\det A=ad-bc.$$ $\det A$ is also denoted by $|A|$ or $\left|\begin{array}{ccc} a & b\\ c & d \end{array}\right|$. In terms of the column … Continue reading

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## Orthogonal Bases

Let $V$ be a vector space with a positive definite scalar product $\langle\ ,\ \rangle$. A basis $\{v_1,\cdots,v_n\}$ of $V$ is said to be orthogonal if $\langle v_i,v_j\rangle=0$ if $i\ne j$. In addition, if $||v_i||=1$ for all $i=1,\cdots,n$, then the … Continue reading