## Linear Algebra and Its Applications, Exercise 2.5.5

Exercise 2.5.5. Given the incidence matrix $A$ from exercise 2.5.1 and the diagonal matrix $C = \begin{bmatrix} c_1&0&0 \\ 0&c_2&0 \\ 0&0&c_3 \end{bmatrix}$

compute $A^TCA$ and show that the 2 by 2 matrix resulting from removing the third row and third column is invertible..

Answer: From exercise 2.5.1 we have the incidence matrix $A = \begin{bmatrix} 1&-1&0 \\ 0&1&-1 \\ 1&0&-1 \end{bmatrix}$

so that $A^TCA = \begin{bmatrix} 1&0&1 \\ -1&1&0 \\ 0&-1&-1 \end{bmatrix} \begin{bmatrix} c_1&0&0 \\ 0&c_2&0 \\ 0&0&c_3 \end{bmatrix} \begin{bmatrix} 1&-1&0 \\ 0&1&-1 \\ 1&0&-1 \end{bmatrix}$ $= \begin{bmatrix} c_1&0&c_3 \\ -c_1&c_2&0 \\ 0&-c_2&-c_3 \end{bmatrix} \begin{bmatrix} 1&-1&0 \\ 0&1&-1 \\ 1&0&-1 \end{bmatrix}$ $= \begin{bmatrix} c_1+c_3&-c_1&-c_3 \\ -c_1&c_1+c_2&-c_2 \\ -c_3&-c_2&c_2+c_3 \end{bmatrix}$

Note that the third row of this matrix is equal to -1 times the sum of the first and second rows, so the rows are linearly dependent and the matrix is singular.

If we remove the third row and third column of $A^TCA$ we obtain the following matrix: $\begin{bmatrix} c_1+c_3&-c_1 \\ -c_1&c_1+c_2 \end{bmatrix}$

This matrix is nonsingular (except for certain values of $c_1$, $c_2$ and $c_3$ as discussed below); its inverse is $1/((c_1+c_3)(c_1+c_2) - (-c_1)(-c_1)) \begin{bmatrix} c_1+c_2&-(-c_1) \\ -(-c_1)&c_1+c_3 \end{bmatrix}$ $= 1/(c_1^2+c_1c_2+c_3c_1+c_3c_2-c_1^2) \begin{bmatrix} c_1+c_2&c_1 \\ c_1&c_1+c_3 \end{bmatrix}$ $1/(c_1c_2+c_1c_3+c_2c_3) \begin{bmatrix} c_1+c_2&c_1 \\ c_1&c_1+c_3 \end{bmatrix}$

Note that if $c_1c_2+c_1c_3+c_2c_3 = 0$ then the 2 by 2 matrix is singular and has no inverse. This would be true, for example, if $c_1 = c_2 = 2$ and $c_3 = -1$ so that the 2 by 2 matrix derived from $A^TCA$ is $\begin{bmatrix} 1&-2 \\ -2&4 \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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