If
𝑀
=
[
1

0

0

0

1

0

6

3

1
]
,
then which of the following is
𝑀
−
1
?

Question

If
𝑀
=
[
1
	
0
	
0

0
	
1
	
0

6
	
3
	
1
]
,
then which of the following is 
𝑀
−
1
?

BlackTom AI · Accepted Answer

Because M is lower triangular with 1’s on the diagonal, its inverse is also lower triangular and can be found by solving M v = e_j for each standard basis vector e_j (or by undoing the row operation that produced M from I).

1. Relevant concepts
- Inverse of a triangular matrix is triangular of the same type.
- To find M^{-1} compute columns v_j satisfying M v_j = e_j.
- Back-substitution works directly because M is lower triangular.

2. Step-by-step solution (compute columns of M^{-1})
Write M and the three right-hand sides:
M = [1 0 0; 0 1 0; 6 3 1].

Column 1: solve M v = e1 = [1,0,0]^T.
Equations:
1) x1 = 1
2) x2 = 0
3) 6 x1 + 3 x2 + 1 x3 = 0
Substitute x1=1, x2=0 into (3): 6(1)+3(0)+x3 = 0 ⇒ x3 = −6.
So first column = [1, 0, −6]^T.

Column 2: solve M v = e2 = [0,1,0]^T.
1) x1 = 0
2) x2 = 1
3) 6 x1 + 3 x2 + x3 = 0 ⇒ 6(0)+3(1)+x3 = 0 ⇒ x3 = −3.
So second column = [0, 1, −3]^T.

Column 3: solve M v = e3 = [0,0,1]^T.
1) x1 = 0
2) x2 = 0
3) 6(0)+3(0)+x3 = 1 ⇒ x3 = 1.
So third column = [0, 0, 1]^T.

3. Assemble M^{-1} by columns:
M^{-1} = [1 0 0; 0 1 0; −6 −3 1].

4. Why this is correct
M was formed from I by adding 6·row1 and 3·row2 to row3; the inverse must subtract those same multiples from row3, yielding the last row (−6, −3, 1). Determinant is 1 so inverse exists and entries match the computed solution.

5. Why other options are wrong
- Option with row2 = [1 1 0] changes a diagonal entry (not consistent with M^{-1}).
- Option with −7 in last row has incorrect subtraction amount for the first column.
- Option with −2 in last row has the wrong value for the (3,2) entry.
Each deviates from the solved linear systems above.

6. Conclusion
The correct inverse is [1 0 0; 0 1 0; −6 −3 1], matching the provided answer.

If 𝑀 = [ 1 0 0 0 1 0 6 3 1 ] , then which of the following is 𝑀 − 1 ?单项选择题

类似问题

If [ 3 − 4 − 4 − 5 ] 𝑋 + [ − 1 2 − 5 − 1 ] = [ 4 1 − 1 3 ] 𝑋 , which of the following matrices is 𝑋 ?

Let 𝐴 = [ 0 0 0 3 5 − 1 − 1 2 3 ] , and 𝐵 = [ 1 − 4 2 3 − 3 3 0 1 − 1 ] . Which of the following options is − 2 𝐴 𝐵 𝑇 ?

True or False: Given an array of features x and an array of corresponding coefficients c, you can compute a predicted value by doing x @ c

Complete this scalar multiplication: The solution will be:

Suppose you are given a random vector ( n x 1 matrix ) where n=9. How many rows are in V(X)? c002.p0180.q003

Suppose 𝐴 is an 𝑛 × 𝑛 invertible matrix. Which of the following statements is FALSE?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单