Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) be such that \(\mathrm{b}(\mathrm{a}+\mathrm{c}) \neq 0\) If \(\left|\begin{array}{ccc}\mathrm{a} & \mathrm{a}+1 & \mathrm{a}-1 \\\ -\mathrm{b} & \mathrm{b}+1 & \mathrm{~b}-1 \\ \mathrm{c} & \mathrm{c}-1 & \mathrm{c}+1\end{array}\right|+\left|\begin{array}{lll}\mathrm{a}+1 & \mathrm{~b}+1 & \mathrm{c}-1 \\ \mathrm{a}-1 & \mathrm{~b}-1 & \mathrm{c}+1 \\\ (-1)^{\mathrm{n}+2} \mathrm{a} & (-1)^{\mathrm{n}+1} \mathrm{~b} & (-1)^{\mathrm{n}} \mathrm{c}\end{array}\right|=0\) then \(\mathrm{n}\) is (a) Zero (b) any even integer (c) any odd integer (d) any integer

Understanding matrix determinants is crucial when it comes to solving systems of linear equations or evaluating the properties of a matrix. Essentially, the determinant is a scalar value that is a function of the entries of a square matrix. It gives important information about the matrix, such as whether there is an inverse, and if used in a system of equations, whether the system has a unique solution.

For example, a determinant of zero indicates that the matrix is not invertible and the system of equations may have no solution or an infinite number of solutions. In the given problem, the determinants of matrices are calculated to find a solution to the equation.

Sarrus' rule is a simple method to calculate the determinant of a 3x3 matrix. This technique involves multiplying diagonal products and summing them, then subtracting the product of the diagonals that go in the reverse direction. It's a shortcut that avoids more complex cofactor expansion methods.

To apply Sarrus' rule, write down the matrix and then copy the first two columns of the matrix to the right. This extended matrix helps you remember which diagonals to multiply. Then, sum the products of the diagonals from the top left to the bottom right and subtract the sum of the products of the diagonals from the bottom left to the top right. It is important to note Sarrus' rule does not apply to matrices larger than 3x3.

The algebraic properties of determinants assist in understanding and computing determinants efficiently. Some fundamental properties include:

• The determinant of a matrix remains the same if its rows and columns are interchanged.
• If two rows (or columns) of a determinant are exactly the same or are proportional, then its value is zero.
• Adding a multiple of one row to another row does not change the value of the determinant.
• The determinant of a product of matrices equals the product of their determinants.
• Multiplying a row or column by a scalar multiplies the determinant by that scalar.

The use of these properties simplifies the calculations in the given problem and confirms that the determinant is a crucial attribute when characterizing a matrix.

A system of linear equations consists of two or more equations that share a common set of variables. The main goal when dealing with a system is to find the values of the variables that satisfy all equations concurrently. Determinants play a vital role in this, particularly in Cramer's rule, which provides an explicit solution to a system with as many equations as unknowns, as long as the determinant of the coefficient matrix is non-zero.

In the context of the given exercise, even though the system of equations is not explicitly provided, the underlying principles are similar. If the determinant of a matrix associated with a system is zero, the system may be dependent or inconsistent, indicating that it might have infinitely many solutions or none. However, non-zero determinants, like in our problem, suggest unique solutions, which are essential for finding values like the variable n.