True or False If A and B are n × n matrices then det(A + B) = det(A) + det(B).trueO false If R is the reduced row echelon form of an m × n matrix A and Ax= b is consistent forsome vector b E R", then Rx = b must also be consistent.truefalse

If A and B are 2 x 2 matrices such that, A=abcd;B=mnpq Then we get, A+B=abcd+mnpq=a+mb+nc+pd+q det…

Answered: If A and B are n x n matrices then…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Determinants

Section3.CM: Cumulative Review

Problem 29CM

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Question

True or False

If A and B are n × n matrices then det(A + B) = det(A) + det(B).
true
O false

If R is the reduced row echelon form of an m × n matrix A and Ax
= b is consistent for
some vector b E R", then Rx = b must also be consistent.
true
false

Expert Solution

Step 1

If A and B are 2 x 2 matrices such that,

$A = (\begin{matrix} a & b \\ c & d \end{matrix}); B = (\begin{matrix} m & n \\ p & q \end{matrix})$

Then we get,

$A + B = (\begin{matrix} a & b \\ c & d \end{matrix}) + (\begin{matrix} m & n \\ p & q \end{matrix}) = (\begin{matrix} a + m & b + n \\ c + p & d + q \end{matrix})$

$d e t A = |A| = |\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c d e t B = |B| = |\begin{matrix} m & n \\ p & q \end{matrix}| = m q - n p$

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