Inverse of a matrix with one value on the diagonal and another value off the diagonal

Question

Let us look at the the matrix

$$ M =\begin{bmatrix} a & b & \dots & b \\ b & a & \dots & b \\ \vdots & b & \ddots & \vdots \\ b & \dots & b & a \end{bmatrix}$$

It has one value $a$ on the main diagonal, and another value $b$ everywhere else. Let us assume that $a \neq b$. I wish to find the inverse of every $n\times n$ matrix of this form ($a$ on the diagonal, $b$ everywhere else).

Just look to the right. It’s the very first related question. — amd, May 28 '17 at 18:11

score 2 · Answer 1 · answered May 28 '17 at 11:51

2

Write $M=(a-b)I+bJ$ where $J$ is the all-one matrix. Try an inverse also of this form $N=xI+yJ$. Taking $I=MN$ will give you two equations in $x$ and $y$ which should be easily soluble.

answered May 28 '17 at 11:51

Angina Seng

158,341

i don't understand your answer.can you briefly tell me the answer.thanks – jadey May 28 '17 at 12:08
What do you get when you expand out $((a-b)I+bJ)(xI+yJ)$? @jadey – Angina Seng May 28 '17 at 12:10
i don't know how to expand this.can you expand this for me,and tell me why you take M=(a-b)I + bJ , N = xI+yJ ? – jadey May 28 '17 at 12:20
@jadey Note that $J^2=nJ$. – egreg May 28 '17 at 13:07
@egreg can you provide me complete answer briefly? – jadey May 28 '17 at 13:18

Inverse of a matrix with one value on the diagonal and another value off the diagonal

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