Use the determinant to find out for which values of the constant λthe matrix A-λIn fails to be i

step 1 Given this matrix A, we're asked to find the values lambda such that the matrix A minus lambda I

Use the determinant to find out for which values of the...

Use the determinant to find out for which values of the constant $\lambda$ the matrix $A-\lambda I_{n}$ fails to be invertible.
$$\left[\begin{array}{lll}
4 & 2 & 0 \\
4 & 6 & 0 \\
5 & 2 & 3
\end{array}\right]$$

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Key Concepts

Eigenvalues

Eigenvalues are the scalars, typically denoted by ?, that satisfy the equation A*v = ?*v for some non-zero vector v. They are found by solving the characteristic polynomial equation. The values of ? for which the matrix A - ?I is non-invertible correspond to the eigenvalues of the matrix A.

Characteristic Polynomial

The characteristic polynomial of a matrix is obtained by computing the determinant of the matrix A - ?I, where I is the identity matrix and ? is a scalar. This polynomial is used to find the eigenvalues of the matrix by solving for ? that satisfy the equation det(A - ?I) = 0.

Invertibility of a Matrix

A matrix is invertible if and only if there exists another matrix that, when multiplied with it, yields the identity matrix. One of the key criteria for invertibility is that the determinant of the matrix must not be zero. When dealing with matrices of the form A - ?I, finding where this matrix is not invertible involves setting its determinant equal to zero.

Determinant

The determinant is a scalar value computed from the elements of a square matrix. It provides important information about the matrix, including whether the matrix is invertible. If the determinant is zero, the matrix is singular (non-invertible), while a non-zero determinant indicates that the matrix is invertible.

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