Meaning and Value of a Determinant

Associated with every square matrix — a matrix having as many rows as columns — is a single real number that captures essential information about the matrix, above all whether a system of linear equations built from it has one definite solution. This number is the determinant of the matrix. Determinants open the Mathematics and Statistics (Commerce) course of the Maharashtra State Board Std XI because they give a quick, purely mechanical way to solve the small systems of linear equations that appear again and again in commercial arithmetic — allocating a fixed budget across products, balancing two costs, or fixing selling prices under given conditions all reduce to a system of linear equations.

For a square matrix of order 2, $A=\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix},$ the determinant is written $|A|$, $\det A$, or $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$, and is defined by cross-multiplying the two diagonals 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 (the leading diagonal), multiply the top-right by the bottom-left (the secondary diagonal), and subtract the second product from the first. For instance, $\begin{vmatrix} 5 & 3 \\ 2 & 4 \end{vmatrix} = 5(4) - 3(2) = 20 - 6 = 14.$

A determinant is a number, not a matrix — the point most often confused when the topic is first met. A matrix is an array of numbers; its determinant is one 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 none, because the very idea depends on there being equally many rows and columns.

Determinant of order three. A determinant of order 3 is formed from a $3\times 3$ matrix and is evaluated by expansion along any one row or column — every choice gives the same final value, which is why one always picks the row or column carrying the most zeros to cut the arithmetic. For $A=\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix},$ expansion along the first row is $|A| = a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}.$

Each term takes one entry of the chosen row, deletes the row and column that entry sits in (leaving a smaller $2\times 2$ determinant, that entry's minor), multiplies the entry by that minor, and attaches a $+$ or $-$ sign that alternates across the row starting with $+$. The fixed sign pattern for a third-order determinant is worth memorising: $\begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}.$

Because every row and column returns the identical value, a determinant can always be re-checked by expanding along a different row or column — precisely the dual-solve discipline every worked example in this chapter follows, since an arithmetic slip in one expansion will not reproduce itself in a second, independent one.