Chapter 3.7, Problem 56HP

(a)

To determine

Determinant of the matrix and zero value of the determinant.

Answer to Problem 56HP

Determinant of the matrix,

  $\text{Det}=ag-fb$

For zero value of the determinant:

  $ag=fb$

Explanation of Solution

Given information:

  $\left|\begin{array}{cc}a& b\\ f& g\end{array}\right|$

We have

  $\left|\begin{array}{cc}a& b\\ f& g\end{array}\right|$

For the determinant of matrix:

  $\text{Det}=ag-fb$

Then

For the 0 value of the determinant:

The value of the determinant is 0 when the product of a andg is equal to the product of f and b .

Such that

  $ag=fb$

(b)

To determine

Values for a , b , f , and g to make the determinant of the coefficient matrix 0 and type of system formed.

Answer to Problem 56HP

Values for a , b , f , and g should be 2, 10, 1, and 5 to make the determinant of the coefficient matrix 0 and system formed is dependent.

Explanation of Solution

Given information:

  $\left|\begin{array}{cc}a& b\\ f& g\end{array}\right|$

We have

  $\left|\begin{array}{cc}a& b\\ f& g\end{array}\right|$

To make the determinant as 0,

The matrix should be

  $\left|\begin{array}{cc}2& 10\\ 1& 5\end{array}\right|$

We know that

Matrix does not represent a system with just one solution.

It is also known

The coefficients of the second row are a factor of the first.

Thus,

The system is dependent because the second row depends on the first row.