The matrix A=[ a b c a b c a b c ] has cigenvalues 0,0,

step 1 For problem number 31, given that the eigenvalues of A are 0 and 0, and Landa 2 is equal to A pl

The matrix A=[ a b c a b c a...

The matrix
$$
A=\left[\begin{array}{lll}
a & b & c \\
a & b & c \\
a & b & c
\end{array}\right]
$$
has cigenvalues $0,0,$ and $a+b+c .$ Determine all values of the constants $a, b,$ and $c$ for which $A$ is nondefective.

Key Concepts

Eigenvalues and Eigenvectors

An eigenvalue of a square matrix is a scalar ? such that there exists a nonzero vector v (an eigenvector) with Av = ?v. Eigenvalues give key insights into the structure of a matrix, and the associated eigenvectors form a basis (or part of one) under certain conditions. In many problems, knowing the eigenvalues simplifies the study of the linear transformation represented by the matrix.

Algebraic and Geometric Multiplicity

The algebraic multiplicity of an eigenvalue is its multiplicity seen as a root of the characteristic polynomial, while the geometric multiplicity is the number of linearly independent eigenvectors corresponding to that eigenvalue. A matrix is diagonalizable (or nondefective) if for every eigenvalue, the geometric multiplicity equals the algebraic multiplicity.

Defective and Diagonalizable Matrices

A defective matrix is one that does not have enough independent eigenvectors to allow for diagonalization; that is, at least one eigenvalue has a smaller geometric multiplicity than its algebraic multiplicity. In contrast, a nondefective matrix is diagonalizable because its eigenvectors form a basis for the space. This concept is central when deciding if a matrix can be simplified via diagonalization.

Rank and Nullity of a Matrix

The rank of a matrix is the dimension of the image (or column space), while the nullity is the dimension of the kernel (or null space). For an n×n matrix, the sum of the rank and nullity equals n. In the case of matrices with repeated rows (or other structure leading to low rank), computing rank and nullity is crucial for determining the number of linearly independent eigenvectors available, particularly for the zero eigenvalue.

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