Her you will learn how to find determinants of matrix 4ร4 with example.
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Determinants of Matrix 4ร4
To evaluate the determinant of a square matrix of order 4 we follow the same procedure as discussed in previous post in evaluating the determinant of a square matrix of order 3.
If A = \(\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}\) is a square matrix of order 4,
then | A | = \(a_{11}\begin{vmatrix} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ย a_{42} & a_{43} & a_{44} \end{vmatrix}\) โ \(a_{12}\begin{vmatrix} a_{21} & a_{23} & a_{24} \\ a_{31} & a_{33} & a_{34} \\ย a_{41} & a_{43} & a_{44} \end{vmatrix}\) + \(a_{13}\begin{vmatrix} a_{21} & a_{22} & a_{24} \\ a_{31} & a_{32} & a_{34} \\ย a_{41} & a_{42} & a_{44} \end{vmatrix}\) โ \(a_{14}\begin{vmatrix} a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ย a_{41} & a_{42} & a_{43} \end{vmatrix}\)
Example :
Find the determinant of A = \(\begin{bmatrix} 1 & 2 & -1ย &ย 3 \\ 2 & 1 & -2ย & 3\\ 3 & 1 & 2 & 1 \\ 1 & -1 & 0 & 2 \end{bmatrix}\).
Solution : | A | = \(\begin{vmatrix} 1 & 2 & -1ย &ย 3 \\ 2 & 1 & -2ย & 3\\ 3 & 1 & 2 & 1 \\ 1 & -1 & 0 & 2 \end{vmatrix}\)
\(\implies\) | A | = \(1\begin{vmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ย -1 & 0 & 2 \end{vmatrix}\) โ \(2\begin{vmatrix} 2 & -2 & 3 \\ 3 & 2 & 1 \\ย 1 & 0 & 2 \end{vmatrix}\) + \((-1)\begin{vmatrix} 2 & 1 & 3 \\ 3 & 1 & 1 \\ย 1 & -1 & 2 \end{vmatrix}\) โ \(3\begin{vmatrix} 2 & 1 & -2 \\ 3 & 1 & 2 \\ย 1 & -1 & 0 \end{vmatrix}\)
| A | =ย (1){\((1)\begin{vmatrix} 2 & 1 \\ย 0 & 2 \end{vmatrix}\) โ \((-2)\begin{vmatrix} 1 & 1 \\ -1 & 2 \end{vmatrix}\) + \((3)\begin{vmatrix} 1 & 2 \\ -1 & 0 \end{vmatrix}\)}
โ (2){\((2)\begin{vmatrix} 2 & 1 \\ย 0 & 2 \end{vmatrix}\) โ \((-2)\begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix}\) + \((3)\begin{vmatrix} 3 & 2 \\ 1 & 0 \end{vmatrix}\)}
+ (-1){\((2)\begin{vmatrix} 1 & 1 \\ย -1 & 2 \end{vmatrix}\) โ \((1)\begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix}\) + \((3)\begin{vmatrix} 3 & 1 \\ 1 & -1 \end{vmatrix}\)}
โ (3){\((2)\begin{vmatrix} 1 & 2 \\ย -1 & 0 \end{vmatrix}\) โ \((1)\begin{vmatrix} 3 & 2 \\ 1 & 0 \end{vmatrix}\) + \((-2)\begin{vmatrix} 3 & 1 \\ 1 & -1 \end{vmatrix}\)}
\(\implies\) | A | = 1{(1)(4 โ 0) โ (-2)(2 + 1) + (3)(0 + 2)} โ 2{(2)(4 โ 0) โ (-2)(6 โ 1) + (3)(0 โ 2)} โ (-1){(2)(2 + 1) โ (1)(6 โ 1) + (3)(-3 โ 1)} โ 3{(2)(0 + 2) โ (1)(0 โ 2) + (-2)(-3 โ 1)}
\(\implies\) | A | = 1(16) โ 2(12) + (-1)(-11) โ 3(14) = -39