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In this lesson, we will learn how to determine the properties of determinants and how to use them to compute the determinant of a matrix.

Q1:

What operation must be performed on the matrix to get the matrix ? How does this operation affect the value of the determinant?

Q2:

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

Q3:

Find the determinant of the matrix

Q4:

Q5:

Determine whether the following statement is true or false: If π΄ β 1 exists, then d e t d e t οΉ π΄ ο = ( π΄ ) β 1 β 1 .

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?

Q7:

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

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.

Q11:

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

Q12:

Q13:

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

Q14:

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

Q15:

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

Q16:

If π΄ is an π Γ π matrix, is it true that d e t d e t ( β π΄ ) = ( β 1 ) ( π΄ ) π ?

Q17:

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

Q18:

Determine whether the following statement is true or false: If π΄ is a real π Γ π matrix, then d e t οΉ π΄ π΄ ο β₯ 0 ο³ .

Q19:

Does the equation d e t d e t d e t ( π΄ + π΅ ) = ( π΄ ) + ( π΅ ) hold for all π Γ π matrices π΄ and π΅ ?

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