Name: How to Find Cofactor Matrix?
Uploaded: 2019-07-19
Duration: 4 min 54 s
Description: A cofactor is a number we get when we remove the column and row of a designated element in a matrix. In this video, we will learn to find the cofactor matrix...

Question

Solve the system of equations by Cramer's rule (if solvable):

x + y + z = 3, a^{2} x + b^{2} y + c^{2} z = a^{2} + b^{2} + c^{2}, a^{3} x + b^{3} y + c^{3} z = a^{3} + b^{3} + c^{3} [a, b, c are unequal and a b + b c + c a \neq 0]

Accepted Answer

We have,

$x + y + z = 3$

$a^{2} x + b^{2} y + c^{2} z = a^{2} + b^{2} + c^{2}$

$a^{3} x + b^{3} y + c^{3} z = a^{3} + b^{3} + c^{3}$

So, $D = |\begin{matrix} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{matrix}|$

We know that If two columns and rows are identical then determinant is zero also if each element of a row(column) of determinant is expressed as a sum of two or more terms then the determinant can be expressed as the sum of two or more determinants.

And $D_{1} = |\begin{array}{ccc} 3 & 1 & 1 \\ a^{2} + b^{2} + c^{2} & b^{2} & c^{2} \\ a^{3} + b^{3} + c^{3} & b^{3} & c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ b^{2} & b^{2} & c^{2} \\ b^{3} & b^{3} & c^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ c^{2} & b^{2} & c^{2} \\ c^{3} & b^{3} & c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}| + 0 + 0$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}|$

$D_{2} = |\begin{array}{cc} 1 & 3 & 1 \\ a^{2} & a^{2} + b^{2} + c^{2} & c^{2} \\ a^{3} & a^{3} + b^{3} + c^{3} & c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & a^{2} & c^{2} \\ a^{3} & a^{3} & c^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & c^{2} & c^{2} \\ a^{3} & c^{3} & c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}|$

And $D_{3} = |\begin{array}{ccc} 1 & 1 & 3 \\ a^{2} & b^{2} & a^{2} + b^{2} + c^{2} \\ a^{3} & b^{3} & a^{3} + b^{3} + c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & b^{2} \\ a^{3} & b^{3} & b^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & b^{2} \\ a^{3} & b^{3} & b^{3} \end{array}| + |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}|$

$= |\begin{array}{ccc} 1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{array}|$

Clearly, $D = D_{1} = D_{2} = D_{3}$

So, $x = \frac{D_{1}}{D} = 1$

$y = \frac{D_{2}}{D} = 1$

$z = \frac{D_{3}}{D} = 1$

Hence, required solution is $x = y = z = 1$ .

Solve the system of equations by Cramer's rule (if solvable): $x + y + z = 3, a^{2} x + b^{2} y + c^{2} z = a^{2} + b^{2} + c^{2}, a^{3} x + b^{3} y + c^{3} z = a^{3} + b^{3} + c^{3} [a, b, c are unequal and a b + b c + c a \neq 0]$

Important Questions on Determinants

Learn and Prepare for any exam you want !

Solve the system of equations by Cramer's rule (if solvable): x+y+z=3, a2x+b2y+c2z=a2+b2+c2, a3x+b3y+c3z=a3+b3+c3 [a,b,c are unequal and ab+bc+ca≠0]

Important Questions on Determinants

Learn and Prepare for any exam you want !

Solve the system of equations by Cramer's rule (if solvable): $x + y + z = 3, a^{2} x + b^{2} y + c^{2} z = a^{2} + b^{2} + c^{2}, a^{3} x + b^{3} y + c^{3} z = a^{3} + b^{3} + c^{3} [a, b, c are unequal and a b + b c + c a \neq 0]$