If the equations a₁x + b₁ = 0 and a₂x + b₂ = 0 are satisfied by the same value of x, what can be concluded about a₁b₂ – a₂b₁?

a₁b₂ – a₂b₁ = 0, indicating that the determinant of the second order is zero.

A determinant of the second order consists of how many rows and columns?

A second-order determinant is written as:

It has 2 rows and 2 columns, forming a 2 × 2 matrix.

Fill in the blank: The expression a₁(b₂c₃ – b₃c₂) + b₁(c₂a₃ – c₃a₂) + c₁(a₂b₃ – a₃b₂) = 0 represents a determinant of the ___ order.

Third

True or False: Determinants can only be formed from square matrices.

True, determinants are defined only for square matrices.

The minor of an element in a determinant is obtained by deleting the ___ and ___ in which the element stands.

row and column

Fill in the blank: A determinant of order three will have ___ minors.

9

Riddle: I am a mathematical tool used to calculate the area, volume, and more. I have rows and columns, but I'm not a table. What am I?

A determinant.

The sum of the products of the elements of any row (or column) of a matrix with their corresponding cofactors always equals the value of the ___ of the matrix.

Determinant

True or False: The sum of the products of the elements of any row with the cofactors of another row is always equal to zero.

True

Fill in the blanks: The determinant can be expressed in six forms, including a₁A₁, b₁B₁, c₁C₁ and ___, ___, ___.

If two rows of a determinant are interchanged, the value of the determinant is ___ in sign.

Consider the 2×2 determinant:

Now, interchange the two rows:

Ans: changed

True or False: If all elements of a row in a determinant are zero, the value of the determinant is non-zero.

False. The value of the determinant is zero.

Fill in the blank: If all elements of a row are multiplied by a constant, the value of the determinant is ___ by that constant.

Multiplied

What happens to the value of a determinant if two rows become identical?

The value of the determinant vanishes, i.e., it equals zero.

What is the result of multiplying two determinants of order three?

The product of the two determinants is equal to the determinant of the product of the two matrices.

If A and B are two 3×3 matrices, then

If D = 0 and at least one of D₁, D₂, D₃ is not zero, what can be concluded about the system of equations?

The equations are inconsistent and have no solution.

Fill in the blank: Cramer's Rule applies to systems of equations with a unique solution when D ≠ ___ and at least one of D₁, D₂, D₃ ≠ 0.

For Cramer's Rule to apply (unique solution), the key condition is:

non-zero

Ans: 0

True or False: Dependent equations in a system always have a unique solution.

False. Dependent equations have infinite solutions.