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Chapter 2: Q 39E (page 98)

Question:Consider a square matrix that differs from the identity matrix at just one entry, off the diagonal, for example,

(100010鈭�1201)

In general, is a matrix M of this form invertible? If so, what is the M-1?

Short Answer

Expert verified

The matrix M is invertible if mij = k (where i鈮�j ), then the ijthentry ofM-1 is -k and all other entries are the same.

Step by step solution

01

Consider the matrix.

Consider the matrix.

M=100010k01

The matrix is said to be invertible if and only if the determinant of the matrix is nonzero.

02

Perform the elementary row operation.

Consider the matrix,

M=100010k01Thereducedmatrixis,M鈭�1=100010鈭�k01

The matrix M is invertible if mij = k (where i鈮�j), then the ijthentry of M-1 is -k and all other entries are the same.

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