91Ó°ÊÓ

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{ll} x & \ln x \\ 1 & 1 / x \end{array}\right|$$

Determine whether the matrix is orthogonal. An invertible square matrix \(A\) is called orthogonal if \(A^{-1}=A^{T}\) $$\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right]$$

Determine whether the points are coplanar $$(1,2,7),(-3,6,6),(4,4,2),(3,3,4)$$

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{rr} x & x \ln x \\ 1 & 1+\ln x \end{array}\right|$$

Find an equation of the plane passing through the three points. $$(1,-2,1),(-1,-1,7),(2,-1,3)$$

Determine whether the matrix is orthogonal. An invertible square matrix \(A\) is called orthogonal if \(A^{-1}=A^{T}\) $$\left[\begin{array}{rr} 1 & -1 \\ -1 & -1 \end{array}\right]$$

The determinant of a \(2 \times 2\) matrix involves two products. The determinant of a \(3 \times 3\) matrix involves six triple products. Show that the determinant of a \(4 \times 4\) matrix involves 24 quadruple products. (In general, the determinant of an \(n \times n\) matrix involves \(n ! n\) -fold products.

Determine whether the matrix is orthogonal. An invertible square matrix \(A\) is called orthogonal if \(A^{-1}=A^{T}\) $$\left[\begin{array}{rr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ -1 / \sqrt{2} & -1 / \sqrt{2} \end{array}\right]$$

Show that the system of linear equations \\[ \begin{array}{l} a x+b y=e \\ c x+d y=f \end{array} \\] has a unique solution if and only if the determinant of the coefficient matrix is nonzero.

Find an equation of the plane passing through the three points. $$(0,-1,0),(1,1,0),(2,1,2)$$