Here you will learn what is determinant of matrix and formula for how to find the determinant of matrix of different order.<\/p>\n
Let’s begin –<\/p>\n
If the equations \\(a_1x + b_1\\) = 0, \\(a_2x + b_2\\) = 0 are satisfied by the same value of x, then \\(a_1b_2 – a_2b_1\\) = 0.\u00a0<\/p>\n
The expression \\(a_1b_2 – a_2b_1\\) is called a determinant of the second order<\/strong>, and it is denoted by<\/p>\n \\(\\begin{vmatrix} a_1 & b_1 \\\\ a_2 & b_2 \\end{vmatrix}\\)\u00a0<\/p>\n<\/blockquote>\n A determinant of second order consists of two rows and two columns.<\/p>\n Next consider the system of equations \\(a_1x + b_1y + c_1\\) = 0, \\(a_2x + b_2y + c_2\\) = 0, \\(a_3x + b_3y + c_3\\) = 0<\/p>\n If these equations are satisfied by the same values of x and y, then on eliminating x and y we get,<\/p>\n \\(a_1(b_2c_3 – b_3c_2)\\) + \\(b_1(c_2a_3 – c_3a_2)\\) + \\(c_1(a_2b_3 – a_3b_2)\\) = 0<\/p>\n The expression on the left is called a determinant of the third order<\/strong>, and is denoted by<\/p>\n \\(\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}\\)\u00a0<\/p>\n<\/blockquote>\n If A = \\([a_1]\\) is a square matrix of order 1, then the determinant of A is defined as<\/p>\n | A | = \\(a_1\\)\u00a0 or,\u00a0 \\(|a_1|\\) = \\(a_1\\)<\/p>\n<\/blockquote>\n If A = \\(\\begin{bmatrix} a_1 & b_1 \\\\ a_2 & b_2 \\end{bmatrix}\\) is a square matrix of order 2,<\/p>\n then the expression \\(a_1b_2 – a_2b_1\\) is defined as the determinant of A.<\/p>\n i.e. | A | = \\(\\begin{vmatrix} a_1 & b_1 \\\\ a_2 & b_2 \\end{vmatrix}\\) = \\(a_1b_2 – a_2b_1\\)<\/p>\n<\/blockquote>\n If A = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) is a square matrix of order 3,<\/p>\n then the expression \\(a_1(b_2c_3 – b_3c_2)\\) – \\(b_1(a_2c_3 – a_3c_2)\\) + \\(c_1(a_2b_3 – a_3b_2)\\) is defined as the determinant of A.<\/p>\n i.e. | A | = \\(\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}\\)\u00a0<\/p>\n = \\(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}\\)<\/p>\n = \\(a_1(b_2c_3 – b_3c_2)\\) – \\(b_1(a_2c_3 – a_3c_2)\\) + \\(c_1(a_2b_3 – a_3b_2)\\)<\/p>\n<\/blockquote>\n\n
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How to Find the Determinant of Matrix<\/h2>\n
Determinant of Matrix of Order 1<\/strong><\/h4>\n
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Determinant of Matrix of Order 2<\/strong><\/h4>\n
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Determinant of Matrix of Order 3<\/strong><\/h4>\n
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\nRelated Questions<\/h3>\n