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Real Matrix with Upper Triangular Block Structure

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** Please see the attached file for the complete problem statement **

Let A be a real matrix having the upper triangular block structure

A_11 A_12 A_13 ... A_1n

0 A_22 A_23 ... A_2n

0 0 A_33 ... A_3n

. . . .
. . . .
. . . .

0 0 0 ... A_nn

where, for all i, j with 1 <= i <= n and 1 <= j <= n, block A_ij is a 2 x 2 matrix; thus A is a 2n x 2n matrix.

(a) Prove that det(A) = det(A_11) * det(A_22) * det(A_33) * ... * det(A_nn).

This is where I need help. I need to know how to write up the proof of this expansion rigorously, but in a reasonable amount of time and space.

(b) Give a simple procedure for computing the eigenvalues of A, including proof.

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Solution Summary

This solution provides a detailed, step-by-step proof for parts (a) and (b), with a full explanation at each step.

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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