Helwan University Faculty Of Engineering: Determinants Chapter II Quiz

9 Questions

What is the value of $x_1$ in the system of equations presented?

$-7$

What is the inverse of matrix A?

$\begin{bmatrix} 0 & 1 & 0 \ 0 & 1 & -\frac{3}{2} \ 0 & -\frac{13}{53} & \frac{10}{53} \end{bmatrix}$

How can the rank of a matrix be determined?

By finding the largest square submatrix with a nonzero determinant

What method can be used to find the rank of a matrix more efficiently than using determinants?

Gaussian Elimination

What is the solution for $x_2$ in the system of equations?

$-3$

What would be the result if you multiplied all elements of matrix A by $-2$?

$\begin{bmatrix} -106 & 26 & 50 \ 38 & -20 & -14 \ -38 & 24 & -24 \end{bmatrix}$

In the system of equations given, what would be the solution for $x_3$?

$0$

What does Cramer's Rule help determine?

$x_1$ Value in Systems of Equations

How can a determinant be used to find the rank of a matrix?

By identifying largest square submatrices with nonzero determinants

Study Notes

Determinants

The determinant of a matrix A is denoted by det(A) or |A|.
For a 2x2 matrix, the determinant is calculated as a11a22 - a12a21.
If the determinant of a matrix is non-zero, the matrix is nonsingular; otherwise, it is singular.

Properties of Determinants

If a matrix has zero rows or columns, its determinant is zero.
Multiplying a row or column by a scalar k multiplies the determinant by k.
Interchanging two rows or columns changes the sign of the determinant.
If two rows of a matrix are the same, the determinant is zero.

Evaluation of Determinants (Cofactor Expansions)

For a 3x3 matrix, the determinant can be calculated using cofactor expansions by a row or column.
The sign rule is + - + for a 3x3 matrix.
Cofactor expansions can be used to find the determinant of a matrix.

Solving Systems of Linear Equations using Determinants (Cramer's Rule)

Cramer's rule is a method for solving systems of linear equations using determinants.
The solution to a system of linear equations can be found by dividing the determinant of a modified matrix by the determinant of the coefficient matrix.

Inverse Matrices and Determinants

The inverse of a matrix can be calculated using determinants.
A matrix has an inverse if and only if its determinant is non-zero.

Rank of a Matrix

The rank of a matrix is the number of rows or columns of the largest square submatrix with a non-zero determinant.
The rank of a 2x2 matrix can be found by checking if the determinant is non-zero.