91Ó°ÊÓ

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

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the given conditions. Solve the system using matrices. $$f(1)=1, f(2)=-1, f(3)=-5$$

Determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, then the determinant of the matrix will be zero.

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the given conditions. Solve the system using matrices. $$f(1)=2, f(2)=9, f(3)=20$$

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the given conditions. Solve the system using matrices. $$f(-2)=-15, f(-1)=7, f(1)=-3$$

Find square matrices \(A\) and \(B\) to demonstrate that \(|A+B| \neq|A|+|B|\).

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the given conditions. Solve the system using matrices. $$f(-2)=-3, f(1)=-3, f(2)=-11$$

Consider square matrices in which the entries are consecutive integers. An example of such a matrix is $$\left[\begin{array}{rrr}4 & 5 & 6 \\ 7 & 8 & 9 \\\ 10 & 11 & 12\end{array}\right]$$ (a) Use a graphing utility to evaluate the determinants of four matrices of this type. Make a conjecture based on the results. (b) Verify your conjecture.

Write a brief paragraph explaining the difference between a square matrix and its determinant.