Calculate the determinants of the following matrices by...

Calculate the determinants of the following matrices by Gaussian elimination: $$ \left[\begin{array}{llll} 1 & 3 & 2 & 1 \\ 0 & 4 & 1 & 6 \\ 0 & 0 & 2 & 1 \\ 1 & 1 & 0 & 4 \end{array}\right],\left[\begin{array}{llll} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array}\right],\left[\begin{array}{llll} 1 & 3 & 2 & 4 \\ 0 & 2 & 1 & 6 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 2 \end{array}\right],\left[\begin{array}{llll} 0 & 0 & 0 & 4 \\ 1 & 2 & 3 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 2 & 6 \end{array}\right] . $$

Calculate the determinants of the following matrices by Gaussian elimination:
$$
\left[\begin{array}{llll}
1 & 3 & 2 & 1 \\
0 & 4 & 1 & 6 \\
0 & 0 & 2 & 1 \\
1 & 1 & 0 & 4
\end{array}\right],\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0
\end{array}\right],\left[\begin{array}{llll}
1 & 3 & 2 & 4 \\
0 & 2 & 1 & 6 \\
0 & 0 & 3 & 0 \\
0 & 0 & 1 & 2
\end{array}\right],\left[\begin{array}{llll}
0 & 0 & 0 & 4 \\
1 & 2 & 3 & 1 \\
0 & 0 & 1 & 1 \\
0 & 1 & 2 & 6
\end{array}\right] .
$$

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is one in which all the entries below the main diagonal are zero. When a matrix is in this form, its determinant is simply the product of the diagonal elements. This property is especially useful because it simplifies the otherwise complex computation of determinants, particularly when obtained through Gaussian elimination.

Elementary Row Operations

Elementary row operations include swapping rows, multiplying a row by a nonzero constant, and adding or subtracting a multiple of one row from another. These operations are essential for transforming a matrix into a simpler form; however, they can affect the determinant in predictable ways, such as changing its sign when rows are swapped or scaling it when a row is multiplied by a constant.

Determinant

A determinant is a scalar value derived from a square matrix that encapsulates important properties such as invertibility and the scaling effect of the linear transformation represented by the matrix. It provides insights into whether a system of equations has a unique solution, no solution, or infinitely many solutions, and also indicates how volumes are scaled under the transformation.

Gaussian Elimination

Gaussian elimination is a systematic method used to simplify a matrix to an upper triangular form through a sequence of row operations. This process makes it easier to compute the determinant and solve systems of linear equations by reducing complex matrices into a form where the main diagonal holds key information.

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