Chapter 6

In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Multiplication of a nonsingular matrix and its inverse is commutative.

Describe how you would explain to another student that the augmented matrix below represents a dependent system of equations. Describe a way to write the infinitely many solutions of this system. \(\left[\begin{array}{rrrrr}1 & -2 & 3 & -6 \\ 0 & 1 & 2 & \vdots & 5 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrrr} 3 & 4 & -2 & 7 \\ 1 & 3 & -1 & 2 \\ 0 & 5 & 7 & 1 \\ 1 & 3 & -1 & 2 \end{array}\right] $$

If \(A\) is a \(2 \times 2\) matrix \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), then \(A\) is invertible if and only if \(a d-b c \neq 0 .\) If \(a d-b c \neq 0\), verify that the inverse is \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\).

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrr} 3 & 2 & -1 \\ -6 & -4 & 2 \\ 5 & -7 & 9 \end{array}\right] $$

Consider matrices of the form $$ A=\left[\begin{array}{cccccc} a_{11} & 0 & 0 & 0 & \cdots & 0 \\ 0 & a_{22} & 0 & 0 & \cdots & 0 \\ 0 & 0 & a_{33} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & a_{n n} \end{array}\right] $$ (a) Write a \(2 \times 2\) matrix and a \(3 \times 3\) matrix of the form of \(A\). Find the inverse of each. (b) Use the result from part (a) to make a conjecture about the inverses of matrices of the form of \(A\).

From 1994 to 2005, the total energy imports \(y\) (in quadrillions of Btu's) to the United States increased in a pattern that was approximately linear (see figure). Find the least squares regression line \(y=a t+b\) for the data shown in the figure by solving the following system using matrices. Let \(t\) represent the year, with \(t=0\) corresponding to 1994 . \(\left\\{\begin{array}{l}12 b+66 a=334.80 \\ 66 b+506 a=1999.91\end{array}\right.\) Use the result to predict the total energy imports in \(2010 .\) Is the estimate reasonable? Explain. (Source: Energy

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrrr} 3 & 0 & 1 & 7 \\ 2 & -1 & 4 & 3 \\ 11 & 5 & -7 & 8 \\ -6 & 3 & -12 & -9 \end{array}\right] $$

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrr} -1 & 3 & 2 \\ 5 & 7 & 0 \\ -1 & 3 & 2 \end{array}\right] $$

In Exercises 91 and 92, determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, the determinant will always be zero.

In Exercises 91 and 92, determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, the determinant of the matrix will be zero.

Evaluate the determinant(s) to verify the equation. \(\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|=-\left|\begin{array}{ll}y & z \\ w & x\end{array}\right|\)

Evaluate the determinant(s) to verify the equation. \(\left|\begin{array}{ll}w & c x \\ y & c z\end{array}\right|=c\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|\)

Evaluate the determinant(s) to verify the equation. \(\left|\begin{array}{rr}w & x \\ c w & c x\end{array}\right|=0\)