91Ó°ÊÓ

Find the volume of the tetrahedron with the given vertices. $$(5,4,-3),(4,-6,-4),(-6,-6,-5),(0,0,10)$$

Guided Proof Prove Property 2 of Theorem 3.3 : When \(B\) is obtained from \(A\) by adding a multiple of a row of \(A\) to another row of \(A, \operatorname{det (B)=\operatorname{det}(A)\) Getting Started: To prove that the determinant of \(B\) is equal to the determinant of \(A,\) you need to show that their respective cofactor expansions are equal. (i) Begin by letting \(B\) be the matrix obtained by adding \(c\) times the \(j\) th row of \(A\) to the \(i\) th row of \(A\) (ii) Find the determinant of \(B\) by expanding in this ith row. (iii) Distribute and then group the terms containing a coefficient of \(c\) and those not containing a coefficient of \(c\) (iv) Show that the sum of the terms not containing a coefficient of \(c\) is the determinant of \(A,\) and the sum of the terms containing a coefficient of \(c\) is equal to \(0 .\)

Determine whether the points are coplanar. $$(1,2,3),(-1,0,1),(0,-2,-5),(2,6,11)$$

Find the values of \(\lambda\) for which the determinant is zero. $$\left|\begin{array}{rrr} \lambda & 0 & 1 \\ 0 & \lambda & 3 \\ 2 & 2 & \lambda-2 \end{array}\right|$$

Determine whether Cramer's Rule is used correctly to solve for the variable. If not, identify the mistake. $$\begin{array}{l}5 x-2 y+z=15 \\\3 x-3 y-z=-7 \\\2 x-y-7 z=-3\end{array}$$ $$x=\frac{\left|\begin{array}{rrr}15 & -2 & 1 \\\\-7 & -3 & -1 \\\\-3 & -1 & -7\end{array}\right|}{\left|\begin{array}{lll}5 & -2 & 1 \\\3 & -3 & -1 \\\2 & -1 & -7\end{array}\right|}$$

Find two \(2 \times 2\) matrices such that \(|A|+|B|=|A+B|\)

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|$$

Proof Prove that if an \(n \times n\) matrix \(A\) is not invertible, then \(A[\operatorname{adj}(A)]\) is the zero matrix.

Prove that the determinant of an invertible matrix \(A\) is equal to \(\pm 1\) when all of the entries of \(A\) and \(A^{-1}\) are integers. Getting Started: Denote det( \(A\) ) as \(x\) and \(\operatorname{det}\left(A^{-1}\right)\) as \(y\) Note that \(x\) and \(y\) are real numbers. To prove that \(\operatorname{det}(A)\) is equal to \(\pm 1,\) you must show that both \(x\) and \(y\) are integers such that their product \(x y\) is equal to 1 (i) Use the property for the determinant of a matrix product to show that \(x y=1\) (ii) Use the definition of a determinant and the fact that the entries of \(A\) and \(A^{-1}\) are integers to show that both \(x=\operatorname{det}(A)\) and \(y=\operatorname{det}\left(A^{-1}\right)\) are integers. (iii) Conclude that \(x=\operatorname{det}(A)\) must be either 1 or \(-1\) because these are the only integer solutions to the equation \(x y=1\)

Prove Theorem 3.9: If \(A\) is a square matrix, then \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) Getting Started: To prove that the determinants of \(A\) and \(A^{T}\) are equal, you need to show that their cofactor expansions are equal. The cofactors are \(\pm\) determinants of smaller matrices, so you need to use mathematical induction. (i) Initial step for induction: If \(A\) is of order \(1,\) then \(A=\left[a_{11}\right]=A^{T}\) so\(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)=a_{11}\) (ii) Assume the inductive hypothesis holds for all matrices of order \(n-1 .\) Let \(A\) be a square matrix of order \(n .\) Write an expression for the determinant of \(A\) by expanding in the first row. (iii) Write an expression for the determinant of \(A^{T}\) by expanding in the first column. (iv) Compare the expansions in (ii) and (iii). The entries of the first row of \(A\) are the same as the entries of the first column of \(A^{T} .\) Compare cofactors (these are the \(\pm\) determinants of smaller matrices that are transposes of one another) and use the inductive hypothesis to conclude that they are equal as well.