Worksheet: Properties of Determinants

In this worksheet, we will practice determining the properties of determinants and using them to compute the determinant of a matrix.

Q1:

Determine whether the following statement is true or false: If 𝐴 𝑥 = 0 for some 𝑥 0 , then d e t ( 𝐴 ) = 0 .

  • Atrue
  • Bfalse

Q2:

Determine whether the following statement is true or false: If 𝐴 1 exists, then d e t d e t 𝐴 = ( 𝐴 ) 1 1 .

  • Atrue
  • Bfalse

Q3:

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

  • Afalse
  • Btrue

Q4:

Does the equation d e t d e t d e t ( 𝐴 + 𝐵 ) = ( 𝐴 ) + ( 𝐵 ) hold for all 𝑛 × 𝑛 matrices 𝐴 and 𝐵 ?

  • Ano
  • Byes

Q5:

Determine whether the following statement is true or false: If 𝐴 is a real 𝑛 × 𝑛 matrix, then d e t 𝐴 𝐴 0 .

  • Atrue
  • Bfalse

Q6:

If 𝐴 is an 𝑛 × 𝑛 matrix, is it true that d e t d e t ( 𝐴 ) = ( 1 ) ( 𝐴 ) 𝑛 ?

  • Ayes
  • Bno

Q7:

Determine whether the statement is true or false: If 𝐴 is an 𝑛 × 𝑛 matrix and 𝐴 = 0 𝑘 for any positive integer, 𝑘 , then d e t ( 𝐴 ) = 0 .

  • Atrue
  • Bfalse

Q8:

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

  • AYes
  • BNo

Q9:

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

  • Afalse
  • Btrue

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:

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

  • A 1 6 𝑚
  • B 4 𝑚
  • C 𝑚
  • D 6 4 𝑚

Q12:

What operation must be performed on the matrix to get the matrix ? 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 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 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.

Q13:

What operation must be performed on the matrix to get the matrix ? 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 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 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.

Q14:

What operation must be performed on the matrix to get the matrix ? 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 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.

Q15:

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

Q16:

Find the determinant of the matrix

Q17:

Using row operations, find the determinant of

Q18:

Find, without expanding, the value of the determinant

Q19:

If the rank of a 4 × 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

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