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Worksheet: Properties of Determinants

Q1:

What operation must be performed on the matrix  π‘Ž 𝑏 𝑐 𝑑  to get the matrix  𝑏 π‘Ž 𝑑 𝑐  ? How does this operation affect the value of the determinant?

  • AThe second matrix is the transpose of the first. The determinant of the second matrix is the same as the determinant of the first matrix.
  • BThe second matrix is the inverse of the first. The determinant of the second matrix is the reciprocal of the determinant of the first matrix.
  • CThe second matrix is the first matrix with the rows swapped. The determinant of the second matrix is βˆ’ 1 times the determinant of the first matrix.
  • DThe second matrix is the first matrix with the columns swapped. The determinant of the second matrix is βˆ’ 1 times the determinant of the first matrix.
  • EThe second matrix is the first matrix with the columns swapped. The determinant of the second matrix is the same as the determinant of the first matrix.

Q2:

Determine whether the following statement is true or false: If 𝐴 π‘₯ = 0 for some π‘₯ β‰  0 , then d e t ( 𝐴 ) = 0 .

  • Atrue
  • Bfalse

Q3:

Find the determinant of the matrix

Q4:

What operation must be performed on the matrix  π‘Ž 𝑏 𝑐 𝑑  to get the matrix  π‘Ž 𝑏 π‘Ž + 𝑐 𝑏 + 𝑑  ? How does this operation affect the value of the determinant?

  • AThe second matrix is the first matrix with the columns swapped. The determinant of the second matrix is βˆ’ 1 times the determinant of the first matrix.
  • BThe second matrix is the inverse of the first. The determinant of the second matrix is the reciprocal of the determinant of the first matrix.
  • CThe second matrix is the first matrix with the rows swapped. The determinant of the second matrix is βˆ’ 1 times the determinant of the first matrix.
  • DThe second matrix is formed by replacing the second row of the first matrix with the sum of its rows. The determinant of the second matrix is the same as the determinant of the first matrix.
  • EThe second matrix is the inverse of the first. The determinant of the second matrix is the same as the determinant of the first matrix.

Q5:

Determine whether the following statement is true or false: If 𝐴 βˆ’ 1 exists, then d e t d e t ο€Ή 𝐴  = ( 𝐴 ) βˆ’ 1 βˆ’ 1 .

  • Atrue
  • Bfalse

Q6:

If the rank of a 4 Γ— 4 matrix is 3, then what, if anything, can be said about the value of its determinant?

  • ANothing without more information
  • BThe determinant equals 1
  • CThe determinant equals 0

Q7:

Determine whether the following statement is true or false: If 𝐴 is an 𝑛 Γ— 𝑛 matrix, then d e t d e t ( 3 𝐴 ) = 3 ( 𝐴 ) .

  • Afalse
  • Btrue

Q8:

Find, without expanding, the value of the determinant

Q9:

Using row operations, find the determinant of

Q10:

Determine whether the following statement is true or false: If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.

  • Atrue
  • Bfalse

Q11:

Determine whether the following statement is true or false: If 𝐴 is a 3 Γ— 3 matrix with a zero determinant, then one column must be a multiple of some other column.

  • Afalse
  • Btrue

Q12:

What operation must be performed on the matrix  π‘Ž 𝑏 𝑐 𝑑  to get the matrix  π‘Ž 𝑏 2 𝑐 2 𝑑  ? How does this operation affect the value of the determinant?

  • AThe second matrix is formed by replacing the second row of the first matrix by the sum of its rows. The determinant of the second matrix is the same as the determinant of the first matrix.
  • BThe second matrix is the inverse of the first. The determinant of the second matrix is the reciprocal of the determinant of the first matrix.
  • CThe second matrix is the first matrix with the rows swapped. The determinant of the second matrix is βˆ’ 1 times the determinant of the first matrix.
  • DThe second matrix is formed by multiplying the second row of the first matrix by 2. The determinant of the second matrix is 2 times the determinant of the first matrix.
  • EThe second matrix is formed by multiplying the second row of the first matrix by 2. The determinant of the second matrix is the same as the determinant of the first matrix.

Q13:

A third-order determinant has value π‘š . Each element of this determinant is multiplied by 4. What is the value of the resulting determinant?

  • A 1 6 π‘š
  • B 4 π‘š
  • C π‘š
  • D 6 4 π‘š

Q14:

Suppose that 𝐴 and 𝐡 are two 𝑛 Γ— 𝑛 matrices whose only difference is that one row of 𝐡 is 4 times the corresponding row of 𝐴 . Is it true that d e t d e t ( 𝐡 ) = 4 ( 𝐴 ) ?

  • Ayes
  • Bno

Q15:

Determine whether the statement is true or false: If 𝐴 is an 𝑛 Γ— 𝑛 matrix and 𝐴 = 0 π‘˜ for any positive integer, π‘˜ , then d e t ( 𝐴 ) = 0 .

  • Atrue
  • Bfalse

Q16:

If 𝐴 is an 𝑛 Γ— 𝑛 matrix, is it true that d e t d e t ( βˆ’ 𝐴 ) = ( βˆ’ 1 ) ( 𝐴 ) 𝑛 ?

  • Ayes
  • Bno

Q17:

An 𝑛 Γ— 𝑛 matrix is called nilpotent if, for any positive integer π‘˜ , it follows that 𝐴 = 0 π‘˜ . If 𝐴 is a nilpotent matrix and π‘˜ is the smallest possible integer such that 𝐴 = 0 π‘˜ , what are the possible values of d e t ( 𝐴 ) ?

Q18:

Determine whether the following statement is true or false: If 𝐴 is a real 𝑛 Γ— 𝑛 matrix, then d e t ο€Ή 𝐴 𝐴  β‰₯ 0  .

  • Atrue
  • Bfalse

Q19:

Does the equation d e t d e t d e t ( 𝐴 + 𝐡 ) = ( 𝐴 ) + ( 𝐡 ) hold for all 𝑛 Γ— 𝑛 matrices 𝐴 and 𝐡 ?

  • Ano
  • Byes