In 4350, use properties of determinants to find the value

D = ` . a b c d

Using Cramers Rule,the value of x thatsatisfiesthe system of equations is x = . ` 2 3 1 -4 ` e 2x + 3y = 5 x - 4y = -3

True or False A determinant can never equal 0

True or False When using Cramers Rule, if , then the system of linear equations is inconsistent.

True or False The value of a determinant remains unchanged if any two rows or any two columns are interchanged.

True or False If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant remains unchanged

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 714, find the value of each determinant.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 1542,solve each system of equations using Cramers Rule if it is applicable. If Cramers Rule is not applicable, say so.

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 4350, use properties of determinants to find the value of each determinant if it is known that 3 x y z u v w 1 2 3 3 = 4

In Problems 5156,solve for x.

In Problems 5156,solve for x.

In Problems 5156,solve for x.

In Problems 5156,solve for x.

In Problems 5156,solve for x.

In Problems 5156,solve for x.

Geometry: Equation of a Line An equation of the line containing the two points and may be expressed as the determinant Prove thisresult by expanding the determinant and comparing the result to the two-point form of the equation of a line.

Geometry: Collinear Points Using the result obtained in Problem 57, show that three distinct points and are collinear (lie on the same line) if and only if x1 y1 1x2 y2 1x3 y3 13 = 0

Geometry: Area of a Triangle A triangle has vertices and .The area of the triangle is given by the absolute value of D, where . Use this formula to find the area of a triangle with vertices (2, 3), (5, 2), and (6, 5). D = 1 2 3 x1 x2 x3 y1 y2 y3 1 1 1

Show that 3 x2 x 1 y2 y 1 z2 z 1 3 = 1y - z21x - y21x - z2.

Complete the proof of Cramers Rule for two equations containing two variables. [Hint: In system (5), page 875, if then and since Now show that equation (6) provides a solution of the system when Then three cases remain:

Interchange columns 1 and 3 of a 3 by 3 determinant. Show that the value of the new determinant is times the value of the original determinant.

Multiply each entry in row 2 of a 3 by 3 determinant by the number Show that the value of the new determinant is k times the value of the original determinant.

Prove that a 3 by 3 determinant in which the entries in column 1 equal those in column 3 has the value 0.

Prove that, if row 2 of a 3 by 3 determinant is multiplied by and the result is added to the entries in row 1, there is no change in the value of the determinant.