Question

If the system of equations \(x+a y=0, a z+y=0, a x+z=0\) has infinite number of solutions then \(\mathrm{a}=\) (a) 0 (b) 1 (c) \(-1\) (d) \(-2\)

Short answer

The correct value of \(a\) for the system of equations to have infinite solutions is \(a=-1\).

Step-by-step solution

Represent the system of equations in matrix form

The given system of equations can be represented as: \[ \left| \begin{array}{c c c} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{array} \right| \] Next, we need to find the determinant of the matrix.

Compute the determinant of the matrix

To find the determinant of the matrix, we first select the first row to find the co-factors and then compute the determinant using cofactor expansion. The determinant of the given matrix is: \[ \begin{vmatrix} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{vmatrix} = 1\begin{vmatrix}1 & a \\ 0 & 1\end{vmatrix} - a\begin{vmatrix}0 & a \\ a & 1\end{vmatrix} \] Simplifying, we get the determinant as: \(Det = 1(1) - a(0 - a^2) = 1 + a^3\)

Determine the condition for infinite solutions

To have infinite solutions, the determinant of the matrix must be zero. Thus, \(Det = 1 + a^3 = 0\) Now, we will find the value of 'a' that satisfies this equation.

Test the given options for 'a'

Now, we will test the given options for 'a' to find which one satisfies the condition for infinite solutions. (a) 0: \(1 + (0)^3 = 1\) (This doesn't satisfy the condition for infinite solutions) (b) 1: \(1 + (1)^3 = 2\) (This doesn't satisfy the condition for infinite solutions) (c) -1: \(1 + (-1)^3 = 0\) (This satisfies the condition for infinite solutions) (d) -2: \(1 + (-2)^3 = -7\) (This doesn't satisfy the condition for infinite solutions) Since option (c) satisfies the condition for infinite solutions, the correct answer is: \[\boxed{a = -1}\]

Key concepts

System of Equations

A system of equations refers to a collection of two or more equations with a common set of variables. Solving such systems means finding the values of the variables that satisfy all the equations at once. These equations can often be linear, meaning that each variable appears with a power of one.

• The system given is:
       1. \(x + ay = 0\)      2. \(az + y = 0\)      3. \(ax + z = 0\)

To solve such a system, we can use algebraic methods or represent it in different formats like matrices, which often simplifies the process. The solutions to the system can be one unique solution, infinitely many solutions, or no solution, depending on certain conditions.

Infinite Solutions

Infinite solutions occur in a system of equations when there are an unlimited number of values for the variables that satisfy all the equations. This is often a result of the equations being dependent on each other. For a linear system represented by matrices, this occurs when the determinant of the corresponding matrix is zero.

• A zero determinant suggests that the equations in the system are dependent.
• This means that one or more of the equations can be derived algebraically from others.

In practical terms, a system with infinite solutions means that we have enough information to express at least one variable in terms of others, but not enough information to pin down specific values for all variables independently.

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, which can be represented by matrices. It is particularly applicable to systems where the determinant of the coefficient matrix is non-zero.

• The rule states that each variable in the system can be expressed as a ratio of determinants.
• Specifically, if \( \det(A) eq 0 \), then each variable can be found by dividing the determinant of a modified matrix (formed by replacing one column of the original coefficient matrix with the constant terms) by the determinant of the coefficient matrix.

However, Cramer's Rule is not applicable when the determinant is zero, as is the case with our given system when \(a = -1\), indicating infinite solutions. Thus, another method needs to be used to explore such systems.

Matrix Representation

Matrix representation is a compact and organized way of expressing systems of equations, making them easier to manipulate and solve using linear algebra techniques. For a system of linear equations:

• Each equation's coefficients become elements of the matrix.
• The matrix size is determined by the number of equations and variables, forming a square matrix in cases with equal numbers of both.

For example, our system \(x + ay = 0\), \(az + y = 0\), and \(ax + z = 0\) can be written as a 3x3 matrix:
\[ \begin{bmatrix} 1 & a & 0 \ 0 & 1 & a \ a & 0 & 1 \end{bmatrix} \] This matrix form allows us to apply various techniques like finding determinants or using methods such as Gaussian elimination more straightforwardly. Such representation also helps in identifying properties like the zero determinant, leading to insights about the system's solutions.