A determinant is a fundamental concept in linear algebra. It is a scalar value that can be calculated from the elements of any square matrix. For a 2x2 matrix, the determinant equals a-d minus b-c. The determinant tells us important properties about the matrix, such as whether it has an inverse.
Let's calculate the determinant of a 2 by 2 matrix step by step. The formula is a-d minus b-c, where a and d are the main diagonal elements, and b and c are the off-diagonal elements. For example, the determinant of the matrix with elements 3, 2, 1, 4 equals 3 times 4 minus 2 times 1, which gives us 12 minus 2, equals 10.
Geometrically, the determinant has a beautiful interpretation. The absolute value of the determinant equals the area of the parallelogram formed by the matrix's column vectors. In this example, vectors v1 and v2 form a parallelogram with area equal to the absolute value of 3 times 2 minus 1 times 1, which is 5. When the determinant is zero, it means the vectors are linearly dependent and lie on the same line.
For a 3 by 3 matrix, calculating the determinant is more complex. We use cofactor expansion along the first row. Each element in the first row is multiplied by its cofactor, which is a 2 by 2 determinant, with alternating signs. For example, this 3 by 3 matrix has determinant equal to 1 times 0 minus 24, minus 2 times 0 minus 20, plus 3 times 0 minus 5, which equals negative 24 plus 40 minus 15, giving us 1.
Determinants have many important properties and applications in mathematics. A key property is that a matrix is invertible if and only if its determinant is non-zero. Other properties include that the determinant of a product equals the product of determinants, and the determinant of a transpose equals the original determinant. Applications include Cramer's rule for solving linear systems, calculating cross products in 3D geometry, and determining areas and volumes in geometric transformations.