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Chapter 2: Q 40E (page 98)

Question:Show that if a square matrix Ahas two equal columns, thenA is not invertible.

Short Answer

Expert verified

When a matrix has equal columns then rrefA鈮�I3, so, the matrix is not invertible.

Step by step solution

01

Consider the matrix

Consider the matrix with two equal columns.

A=abbcddeff

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

The matrix is said to be invertible, if and only if, rref(A)=I3.

02

Perform the elementary row operation

Consider the matrix,

A=abbcddeff

The reduced matrix is,

abbcddeff=cdd0cf鈭�edccf鈭�edc000=100011000鈮�I3

As rrefA鈮�I3, so, the matrix is non-invertible.

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Most popular questions from this chapter

If A is any invertiblen脳n matrix, then A commutes with A-1.

The formula AB=BAholds for alln脳n matrices A and B .

IfAB=lnfor twon脳nmatrices Aand B, then Amust be the inverse of B.

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises16through 23. Show the effect of these transformations on the letter L considered in Example5. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise13

[1000]

Find the matrices of the transformations Tand Ldefined in Exercise 80.
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