The value of 4(λ1+λ2+λ3) where, λ1, λ2, λ3 are the values of λfor which the system of equations

step 1 So in the given question we are told that the value of four times 4 times lambda 1 plus lambda 2

The value of 4(λ1+λ2+λ3) where, λ1, λ2, λ3 are the values of...

The value of $4\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)$ where, $\lambda_{1,} \lambda_{2}, \lambda_{3}$ are the values of $\lambda$ for which the system of equations $(3-\lambda) x+y+$ $4 \mathrm{z}=0 ;(2-\lambda) \mathrm{y}+6 \mathrm{z}=0 ;(5-\lambda) \mathrm{z}=0$ has non trivial solutions, is (a) 40 (b) 10 (c) 4 (d) 400

Key Concepts

Homogeneous Systems

A homogeneous system of linear equations is one in which all the constant terms are zero. Such systems always have the trivial solution (all variables equal to zero), but they may also have nontrivial solutions if the system is dependent. The key point is that nontrivial solutions exist when the coefficient matrix of the system is singular.

Determinant Condition for Nontrivial Solutions

For a homogeneous system, nontrivial solutions exist if and only if the determinant of the coefficient matrix is zero. This determinant condition is essential because it characterizes when the system of equations is linearly dependent, thus allowing for the existence of solutions other than the trivial zero solution.

Triangular Matrices

A triangular matrix (either upper or lower triangular) is a square matrix where all the entries either above or below the main diagonal are zero. One of the important features of triangular matrices is that their determinant is simply the product of the diagonal entries. Moreover, the eigenvalues of a triangular matrix are exactly the entries on its main diagonal.

Trace and Sum of Eigenvalues

The trace of a square matrix is defined as the sum of its diagonal entries. In linear algebra, an important property is that the sum of the eigenvalues of a matrix, counted with their algebraic multiplicities, is equal to the trace of that matrix. This relationship can be useful when dealing with eigenvalue-related problems, such as finding combinations or sums of eigenvalues.

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