Worksheet: Inverse and Power of Diagonal Matrices

In this worksheet, we will practice finding the inverse of a diagonal matrix and determining the value of a diagonal matrix raised to a high power.

Q1:

Find 𝐴 for the matrix 𝐴=200030005 using elementary row operations.

  • A15000100006
  • B⎑⎒⎒⎒⎒⎒⎣21500031000056⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎒⎒⎒⎣11500011000016⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎒⎒⎒⎣120001300015⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎒⎒⎒⎣130000130000130⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦

Q2:

Given that 𝐴=200050003, find 𝐴.

  • AβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’115000βˆ’160π‘Ž00βˆ’110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎒⎒⎒⎣11500016000110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • Cο˜βˆ’2000βˆ’5000βˆ’3
  • DβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’12000βˆ’15000βˆ’13⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎒⎒⎒⎣120001500013⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦

Q3:

Given that 𝐴=400020005, find 𝐴οŠͺ.

  • A25600016000625
  • B16000800020
  • C256161681682020625
  • D640008000125
  • E16000400025

Q4:

Given the matrix 𝐴=ο˜βˆ’2000βˆ’5000βˆ’3, find 𝐴οŠͺ.

  • AβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’116000βˆ’1625000βˆ’181⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • Bο˜βˆ’16000βˆ’625000βˆ’81
  • C⎑⎒⎒⎒⎒⎒⎣1160001625000181⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’150,625000βˆ’11,296000βˆ’110,000⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • E1600062500081

Q5:

Find the result of raising the matrix 1003 to the power of 3.

  • A1009
  • B9001
  • C3006
  • D3009
  • E1006

Q6:

Given the matrix 𝐴=ο”βˆ’300βˆ’7, find 𝐴.

  • Aο”βˆ’2700βˆ’343
  • B⎑⎒⎒⎣134300βˆ’127⎀βŽ₯βŽ₯⎦
  • C2700343
  • DβŽ‘βŽ’βŽ’βŽ£βˆ’12700βˆ’1343⎀βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎣12700βˆ’1343⎀βŽ₯βŽ₯⎦

Q7:

Fill in the blank: If 𝐴 is a diagonal matrix of size 𝑛×𝑛 that has no inverse, then .

  • Amatrix 𝐴 has at least one zero diagonal element
  • Bmatrix 𝐴 has at least two diagonal elements that are multiplicative inverses of each other
  • Cmatrix 𝐴 has at least two diagonal elements that are additive inverses of each other
  • Dmatrix 𝐴 has identical diagonal elements

Q8:

Fill in the blank: If 𝐴=1004, then 𝐴=.

  • Aο˜βˆ’1400βˆ’1
  • Bο˜βˆ’100βˆ’14
  • Cο”βˆ’100βˆ’4
  • D10014
  • E14001

Q9:

If matrix 𝐴 is diagonal and it has a zero diagonal element, is it invertible?

  • AYes
  • BNo

Q10:

Fill in the blank: If 𝐴=ο”π‘Ž00𝑏, then 𝐴=.

  • Aο”π‘Žπ‘›00𝑏𝑛
  • Bο”π‘Ž00π‘ο οŠοŠ
  • Cο”π‘Žπ‘›π‘›π‘ο οŠοŠ
  • Dο”π‘Ž+𝑛00𝑏+𝑛
  • Eο“π‘Žπ‘›π‘›π‘οŸ

Q11:

Given 𝐴=0006, find 𝐴.

  • A36000
  • B60036
  • C360036
  • D00036
  • E66636

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