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Useful properties of determinants that allow simplification.

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Determinants exhibit some properties that can usefully be thought of as \"Rules\":

Rule 1: If two rows (or two columns) of a determinant are interchanged then the value of the determinant is multiplied by ($−1$).

Rule 2: The determinant of a matrix $A$ and the determinant of its transpose $A^T$ are equal.

Rule 3: If two rows (or two columns) of a matrix $A$ are equal then it has zero determinant.

Rule 4: If the elements of one row (or one column) of a determinant are multiplied by $k$, then the determinant is also multiplied by $k$.

Rule 5: If we add (or subtract) a multiple of one row (or column) to another, the value of the determinant is unchanged.

Rule 6: The determinant of a lower triangular matrix, an upper triangular matrix or a diagonal matrix is the product of the elements on the leading diagonal.

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We are asked to give the determinants for various matrices without explicitly calculating them. We can do this by applying the \"rules\" given:

$\\large \\left| \\begin{array}{ccc} \\var{a22} & \\var{a11} & \\var{a21}\\\\ \\var{a11} & -7 & \\var{a11} \\\\\\var{a22}& \\var{a11} & \\var{a21} \\end{array} \\right| =0$

Row $1$ and Row $3$ are equal so the determinant is $0$ (by Rule 3).

$\\large \\left| \\begin{array}{ccc} \\var{a11} & 0 & 0 & 0\\\\ \\var{a12} & \\var{a11} & 0 & 0 \\\\ \\var{a22}& \\var{a21}& 1 & 0 \\\\ \\var{a12} & 0 & \\var{a21} & \\var{a22} \\end{array} \\right| =\\simplify{{a11}{a11}{a22}}$

Calculating the determinant of matrices larger than $3 \\times 3$ can be a laborious process. However, this matrix is triangular so its determinant is merely the product of the values in the leading diagonal (by Rule 6).

If $\\large \\left| \\begin{array}{ccc} \\var{a11} & \\var{a12} \\\\ \\var{a21}& \\var{a22} \\end{array} \\right| =\\var{detA1}$ then $\\large \\left| \\begin{array}{ccc} \\var{a21} & \\var{a22} \\\\ \\var{a11}& \\var{a12} \\end{array} \\right| =\\var{detA2}$

In this case, the second matrix has the same elements as the first but Row $1$ and Row $2$ have been interchanged. So the determinant of the second will be the same as the determinant of the first but $\\times -1$ (by Rule 1).

If $\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} & \\var{B[0][2]}\\\\ \\var{B[1][0]} & \\var{B[1][1]} & \\var{B[1][2]}\\\\ \\var{B[2][0]} & \\var{B[2][1]}& \\var{B[2][2]}\\end{array}\\right|=\\var{detB}$ then $\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[1][0]}& \\var{B[2][0]}\\\\ \\var{B[0][1]} & \\var{B[1][1]}& \\var{B[2][1]}\\\\ \\var{B[0][2]} & \\var{B[1][2]}& \\var{B[2][2]}\\end{array}\\right|=\\var{detB}$

You should, hopefully, be able to see that the second matrix is the transpose of the first. Hence their determinants are equal (by Rule 2).

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By using the properties of determinants (above), that is without calculating any determinants, answer the following:

$\\large \\left| \\begin{array}{ccc} \\var{a22} & \\var{a11} & \\var{a21}\\\\ \\var{a11} & -7 & \\var{a11} \\\\\\var{a22}& \\var{a11} & \\var{a21} \\end{array} \\right| =$ [[0]]

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