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Determinant of a Matrix


In this section we study the determinant of an \(n\times n\) matrix \(A=[a_{ij}]\), denoted by \(\det(A)\) or \(\det A\) or \(|A|\) or \[\left| \begin{array}{cccc} a_{11}&a_{12}&\cdots &a_{1n}\\ a_{21}&a_{22}&\cdots &a_{2n}\\ \vdots&\vdots& \ddots &\vdots\\ a_{m1}&a_{m2}&\cdots &a_{mn} \end{array} \right|.\]

To define \(\det(A)\) recursively, we denote \(A(i,j)\) for the the matrix obtained from \(A\) by deleting row \(i\) and column \(j\) of \(A\).

Definition. If \(A=[a_{11}]\), then \(\det(A)=a_{11}\). If \(A=\left[\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right]\), then \(\det(A)=a_{11}a_{22}-a_{12}a_{21}\). For an \(n\times n\) matrix \(A=[a_{ij}]\) where \(n\geq 3\), \[\det(A)=\sum_{i=1}^n (-1)^{1+i} a_{1i} \det A(1,i)=a_{11} \det A(1,1)-a_{12}\det A(1,2)+\cdots+(-1)^{n+1} a_{1n} \det A(1,n).\]

Example. We find \(\det(A)\) for \(A=\left[\begin{array}{rrr} 1&2&3\\ 1&3&5\\ 1&4&2\end{array} \right]\). \[\begin{align*} \det(A) &=a_{11} \det A(1,1)-a_{12}\det A(1,2)+a_{13} \det A(1,3)\\ &=1 \left| \begin{array}{rr}3&5\\4&2\end{array} \right| -2 \left| \begin{array}{rrr}1&5\\1&2\end{array} \right| +3 \left| \begin{array}{rr}1&3\\1&4\end{array} \right| \\ &=1(3\cdot 2-5\cdot 4)-2(1\cdot 2-5\cdot 1) +3(1\cdot 4-3\cdot 1)\\ &=-5 \end{align*}\]

Definition. For an \(n\times n\) matrix \(A=[a_{ij}]\) where \(n\geq 2\), the \((i,j)\) minor, denoted by \(m_{ij}\), is \(m_{ij}=\det A(i,j)\) and the \((i,j)\) cofactor, denoted by \(c_{ij}\), is \[c_{ij}=(-1)^{i+j} m_{ij} =(-1)^{i+j}\det A(i,j).\]

Remark. We defined \(\det(A)\) as the cofactor expansion along the first row of \(A\): \[\det(A)=\sum_{i=1}^n (-1)^{1+i}a_{1i} \det A(1,i)= \sum_{i=1}^n a_{1i} c_{1i}.\] But it can be proved that \(\det(A)\) is the cofactor expansion along any row or column of \(A\).

Theorem. Let \(A\) be an \(n\times n\) matrix. Then for each \(i,j=1,2,\ldots,n\), \[\det(A)= \sum_{j=1}^n a_{ij} c_{ij} = \sum_{i=1}^n a_{ij} c_{ij} .\]

The preceding theorem can be proved using the following equivalent definition of determinant: \[\det(A)=\sum_{\sigma \in S_n} \left( \operatorname{sign}(\sigma) \prod_{i=1}^n a_{i\sigma(i)} \right),\] where \(\sigma\) runs over all \(n!\) permutations \(\sigma\) of \(\{1,2,\ldots,n\}\). (This requires study of permutations)

Corollary. Let \(A=[a_{ij}]\) be an \(n\times n\) matrix.

  1. \(\det(A^T)=\det(A)\).

  2. If \(A\) is a triangular matrix, then \(\det(A)=a_{11}a_{22}\cdots a_{nn}\).

(a) Note that the \((i,j)\) cofactor of \(A\) is the \((j,i)\) cofactor of \(A^T\). The cofactor expansions along the first rows to get \(\det(A)\) would be same as cofactor expansions along the first columns to get \(\det(A^T)\).

(b) If \(A\) is an upper-triangular matrix, then by cofactor expansions along the first rows we get \(\det(A)=a_{11}a_{22}\cdots a_{nn}\). Similarly if \(A\) is a lower-triangular matrix, then by cofactor expansions along the first columns we get \(\det(A)=a_{11}a_{22}\cdots a_{nn}\).

Example. \(A=\left[\begin{array}{rrrrr} 1&2&3&4&5\\ 3&0&1&3&2\\ 0&0&4&3&0\\ 0&0&0&2&1\\ 2&0&0&0&3 \end{array} \right].\) We compute \(\det(A)\) using rows or columns with maximum number of zeros at a step. So first we choose column 2 and do cofactor expansion along it: \[\det(A)=-2 \left|\begin{array}{rrrr} 3&1&3&2\\ 0&4&3&0\\ 0&0&2&1\\ 2&0&0&3 \end{array} \right|\] Now we have 5 choices: row 2,3,4 and column 1,2. We do cofactor expansion along row 4: \[\det(A)=-2 \left( -2 \left|\begin{array}{rrr} 1&3&2\\ 4&3&0\\ 0&2&1 \end{array} \right| +3 \left|\begin{array}{rrr} 3&1&3\\ 0&4&3\\ 0&0&2 \end{array} \right| \right) \] Since the second determinant is a determinant of an upper-triangular matrix, its determinant is \(3\cdot 4 \cdot 2=24\). We do cofactor expansion along column 3 for the first determinant. \[\begin{align*} \det(A) &=-2 \left( -2 \left( 2 \left|\begin{array}{rr} 4&3\\ 0&2\end{array} \right| +1 \left|\begin{array}{rrrr} 1&3\\ 4&3 \end{array} \right| \right)+3\cdot 24 \right)\\ &= -2 \left( -2 \left(2(4\cdot 2-0) +1 (1\cdot 3-3\cdot 4)\right)+72 \right)\\ &=-116 \end{align*}\]

Some applications of determinants:

  1. Determinant as volume: Suppose a hypersolid \(S\) in \(\mathbb R^n\) is given by \(n\) concurrent edges that are represented by column vectors of an \(n\times n\) matrix \(A\). Then the volume of \(S\) is \(|\det(A)|\).

    Example. Let \(\overrightarrow{r_1}=[a_1,b_1,c_1]^T\), \(\overrightarrow{r_2}=[a_2,b_2,c_2]^T\), \(\overrightarrow{r_3}=[a_3,b_3,c_3]^T\). \(A=[\overrightarrow{r_1}\;\overrightarrow{r_2}\;\overrightarrow{r_3}]=\left[\begin{array}{ccc}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{array}\right]\) and the volume of the parallelepiped with concurrent edges given by \(\overrightarrow{r_1},\overrightarrow{r_2},\overrightarrow{r_3}\) is \[|\det(A)|=|a_1(b_2c_3-b_3c_2)-a_2(b_1c_3-b_3c_1)+a_3(b_1c_2-b_2c_1)|.\]

  2. Equation of a plane: Consider the plane passing through three distinct points \(P_1(x_1,y_1,z_1)\), \(P_2(x_2,y_2,z_2)\), and \(P_3(x_3,y_3,z_3)\). Let \(P(x,y,z)\) be a point on the plane. So the volume of the parallelepiped with concurrent edges \(\overrightarrow{P_1P}\), \(\overrightarrow{P_2P}\), and \(\overrightarrow{P_3P}\) is zero. \[\left|\begin{array}{ccc}x-x_1&x-x_2&x-x_3\\y-y_1&y-y_2&y-y_3\\z-z_1&z-z_2&z-z_3\end{array}\right|=0.\]

  3. Volume after transformation: Let \(T:\mathbb R^n\to \mathbb R^n\) be a linear transformation with the standard matrix \(A\). Let \(S\) be a bounded hypersolid in \(\mathbb R^n\). Then the volume of \(T(S)\) is \(|\det(A)|\) times the volume of \(S\).

    Example. Let \(A=\left[\begin{array}{cc}a&0\\0&b\end{array}\right]\) and \(D=\{(x,y)\;|\;x^2+y^2\leq 1\}\). Consider \(T:\mathbb R^2\to \mathbb R^2\) defined by \(T([x, y]^T)=A[x, y]^T\). Note \(T(D)=\{(x,y)\;|\;\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\}\). So the area of ellipse = the area of \(T(D)=\det(A)\cdot A(D)=ab\cdot \pi 1^2=\pi ab\).

  4. Change of variables: Suppose variables \(x_1,\ldots,x_n\) are changed to \(v_1,\ldots,v_n\) by \(n\) differentiable functions \(f_1,\ldots,f_n\) so that \[\begin{eqnarray*} v_1&=&f_1(x_1,\ldots,x_n)\\ v_2&=&f_2(x_1,\ldots,x_n)\\ &\vdots &\\ v_n&=&f_n(x_1,\ldots,x_n). \end{eqnarray*}\] So we have a function \(F:\mathbb R^n\to \mathbb R^n\) defined by \[F(x_1,\ldots,x_n)=(f_1(x_1,\ldots,x_n),\ldots,f_n(x_1,\ldots,x_n)).\] The Jacobian matrix of \(F:\mathbb R^n\to \mathbb R^n\) is the following \[\frac{\partial(f_1,\ldots,f_n)}{\partial(x_1,\ldots,x_n)}= \left[\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}&\cdots&\frac{\partial f_1}{\partial x_n}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_n}{\partial x_1}&\cdots&\frac{\partial f_n}{\partial x_n} \end{array} \right].\] The change of variables formula for integrals is \[\int_{F(U)}G(\overrightarrow{v})d\overrightarrow{v}= \int_{U}G(\overrightarrow{x})\left|\frac{\partial(f_1,\ldots,f_n)}{\partial(x_1,\ldots,x_n)}\right| d\overrightarrow{x}.\]

    Example. So \((x,y)=F(r,\theta)=(ar\cos\theta,br\sin\theta)\) and \(F([0,1]\times[0,2\pi])\) is the region inscribed by the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). The Jacobian matrix is \[\frac{\partial(x,y)}{\partial(r,\theta)}= \left[\begin{array}{cc} \frac{\partial x}{\partial r}&\frac{\partial x}{\partial \theta}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial \theta} \end{array} \right]= \left[\begin{array}{cc} a\cos\theta&-ar\sin\theta\\ b\sin\theta&br\cos\theta \end{array} \right]\text{ and } \left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|=abr.\] By the change of variables formula, \[\int_{F([0,1]\times[0,2\pi])}1\;d\overrightarrow{v}= \int_{\theta=0}^{2\pi}\int_{r=0}^11\; \left|\frac{\partial(x,y)}{\partial(r,\theta)}\right| drd\theta=ab\cdot \pi.\]

  5. Wronskian: The Wroskian of \(n\) real-values differentiable functions \(f_1,\ldots,f_n\) is \[W(f_1,\ldots,f_n)(x)= \left|\begin{array}{ccc} f_1(x)&\cdots&f_n(x)\\ f^{'}_1(x)&\cdots&f^{'}_n(x)\\ \vdots&\ddots&\vdots\\ f^{(n-1)}_1(x)&\cdots&f^{(n-1)}_n(x)\\ \end{array} \right|.\] \(f_1,\ldots,f_n\) are linearly independent functions iff \(W(f_1,\ldots,f_n)\) is not identically zero.

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