Meaning of a Determinant

Every square matrix — one with the same number of rows as columns — has a single number associated with it that summarises important information about the matrix, such as whether the system of linear equations built from that matrix has a unique solution. This number is called the determinant of the matrix. Determinants are a core topic of the Odisha CHSE Std 11 Business Mathematics and Statistics syllabus, because they give a fast, mechanical way to solve the systems of linear equations that occur throughout business problems — costing, break-even analysis, and allocation problems all reduce, sooner or later, to a system of linear equations.

For a square matrix $A$ of order 2, $A=\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}$ the determinant of $A$ is written $|A|$ or $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$, and is defined as the number obtained by cross-multiplying and subtracting: $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - b_1 a_2.$

In words: multiply the top-left entry by the bottom-right entry, multiply the top-right entry by the bottom-left entry, and subtract the second product from the first. For example, $\begin{vmatrix} 4 & 3 \\ 2 & 5 \end{vmatrix} = 4(5) - 3(2) = 20 - 6 = 14.$

A determinant is a number, not a matrix — this is the point students most often confuse when they first meet the topic. A matrix is a rectangular (here, square) array of numbers; its determinant is a single real number computed from that array by the rule above. Only a square matrix has a determinant at all — a $2\times 3$ or $3\times 2$ matrix has no determinant defined for it, because the very idea of a determinant depends on the matrix having equally many rows and columns.

The determinant is only zero when the two rows (equivalently, the two columns) of the matrix are proportional to each other — a fact that becomes important later in the chapter when a zero determinant signals that a system of equations has no unique solution, or that three given points lie on a single straight line.