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Worksheet: Scalar Multiplication of Matrices

Q1:

Fill in the blank. Columns of an 𝑛 Γ— 𝑛 matrix 𝐴 are an orthonormal basis for β„‚ 𝑛 , if and only if 𝐴 is a matrix.

  • Asymmetric
  • Bnormal
  • Csquare
  • Dunitary

Q2:

Given the matrix what is the greatest number π‘˜ for which no entry of π‘˜ 𝐴 is greater than 1?

  • A βˆ’ 1 1 6
  • B 1 1 6
  • C βˆ’ 1 7
  • D 1 7
  • E 1 2 5

Q3:

Let 𝑍 be a 2 Γ— 3 matrix whose entries are all zero. If 𝐴 is any 2 Γ— 3 matrix, which of following is equivalent to 5 𝐴 βˆ’ 3 𝑍 ?

  • A βˆ’ 2 𝐴 𝑍
  • B 2 𝑍 𝐴
  • C βˆ’ 3 𝑍
  • D 5 𝐴
  • E 2 𝐴

Q4:

Given that find the value of .

Q5:

Given that , find .

  • A
  • B
  • C
  • D

Q6:

Using the properties of determinants, find the value of

Q7:

Let 𝐴 =  1 1 1 0 βˆ’ 2  and 𝐼 be the 2 Γ— 2 identity matrix. Find 𝐴 βˆ’ 3 𝐼 , 𝐴 + 4 𝐼 , and their product ( 𝐴 βˆ’ 3 𝐼 ) ( 𝐴 + 4 𝐼 ) , and then use this to express 𝐴 2 as a combination of 𝐴 and 𝐼 .

  • A  4 1 1 0 1  ,  βˆ’ 3 1 1 0 βˆ’ 6  ,  0 0 0 0  , 𝐴 = 𝐴 + 1 2 𝐼 2 .
  • B  0 1 1 0 βˆ’ 3  ,  5 1 1 0 2  ,  1 0 2 2 0 4  , 𝐴 = ( βˆ’ 1 ) 𝐴 + 1 2 𝐼 2 .
  • C  βˆ’ 2 1 1 0 βˆ’ 5  ,  5 1 1 0 2  ,  1 2 7 7 0 βˆ’ 2  , 𝐴 = 7 𝐴 + 1 2 𝐼 2 .
  • D  βˆ’ 2 1 1 0 βˆ’ 5  ,  5 1 1 0 2  ,  0 0 0 0  , 𝐴 = ( βˆ’ 1 ) 𝐴 + 1 2 𝐼 2 .
  • E  4 1 1 0 1  ,  βˆ’ 3 1 1 0 βˆ’ 6  ,  βˆ’ 1 1 1 1 0 βˆ’ 1 4  , 𝐴 = 𝐴 + 1 2 𝐼 2 .

Q8:

Find numbers π‘Ž , 𝑏 , and 𝑐 so that π‘Ž  1 1 0 βˆ’ 1  + 𝑏  1 0 0 1  + 𝑐  0 βˆ’ 1 1 0  =  1 0 βˆ’ 1 3  .

  • A π‘Ž = 1 , 𝑏 = 3 , 𝑐 = 1
  • B π‘Ž = βˆ’ 1 , 𝑏 = 3 , 𝑐 = βˆ’ 1
  • C π‘Ž = 1 , 𝑏 = βˆ’ 2 , 𝑐 = 1
  • D π‘Ž = βˆ’ 1 , 𝑏 = 2 , 𝑐 = βˆ’ 1
  • E π‘Ž = 1 , 𝑏 = 2 , 𝑐 = 1