THE DETERMINANT OF A MATRIX

Definitions

- Square Matrix: A matrix whose number of columns is equivalent to its number of rows. (n x n)

- Permutation: There is a set of integers (1,2,3...n) arranged in ascending order. The rearrangement of these elements is called the permutation.

- Determinant: For square matrices, the determinant (det) is defined by the sum of all the permutations of the set (1,2,3...n). The sign for each integer is negative if the permutation is even and positive if the permutation is odd.

det(A) = $ sum((+ or -)a1j1 * (+ or -)a2j2... (+ or -)anjn) $

  >  The determinant is used for determining whether the matrix is nonsingular or not and, therefore, can tell you if there
     is a unique solution or not to the linear combination of the matrix.  The determinant is also used as on number to
     represent the whole array of numbers that make up the matrix.



How to Find the Determinant of...

2 x 2 Matrices

A = $ \left[\begin{array}{cccc}a&b\\c&d\end{array}\right] $

Begin by multiplying the value in the first row and first column by the value in the second row and second column. Subtract this number by the number in the first row and the second column multiplied by the number in the second row and first column.

det(A) = a*d - b*c




3 x 3 Matrices

A = $ \left[\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] $

Begin by selecting the number in the first row and first column. Multiply this number by the value of the determinant of the 2 x 2 matrix created when you omit the firs row and first column.

 a*det($ \left[\begin{array}{cccc}e&f\\h&i\end{array}\right] $)

Then, from this number, you subtract the number of the first row, second column multiplied by the 2 x 2 matrix created when you omit the first row and second colmumn.

 a*det($ \left[\begin{array}{cccc}e&f\\h&i\end{array}\right] $) - b*det($ \left[\begin{array}{cccc}d&f\\g&i\end{array}\right] $)

Finally, you add the number from the first row and the third column multiplied by the determinant created when you omit the first row and the third column.

 a*det($ \left[\begin{array}{cccc}e&f\\h&i\end{array}\right] $) - b*det($ \left[\begin{array}{cccc}d&f\\g&i\end{array}\right] $) + c*det($ \left[\begin{array}{cccc}d&e\\g&h\end{array}\right] $)

Therefore...

det(A) = a*(e*i - f*h) - b*(d*i - f*g) + c*(d*h - e*g)




4 x 4 Matrices

A = $ \left[\begin{array}{cccc}a&b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p\end{array}\right] $

Begin by multiplying the number in the first row and first column of the matrix by the determinant of the 3 x 3 matrix created when you omit the first row and first column.

 a*det($ \left[\begin{array}{cccc}f&g&h\\j&k&l\\n&o&p\end{array}\right] $)

Next, subtract this by the number in the first row and second column multiplied by the determinant of the 3 x 3 matrix created when you omit the first row and second column.

 a*det($ \left[\begin{array}{cccc}f&g&h\\j&k&l\\n&o&p\end{array}\right] $) - b*det($ \left[\begin{array}{cccc}e&g&h\\i&k&l\\m&o&p\end{array}\right] $)

Then, add this by the number in the first row and third column multiplied by the determinant of the 3 x 3 matrix created when you omit the first row and third column.

 a*det($ \left[\begin{array}{cccc}f&g&h\\j&k&l\\n&o&p\end{array}\right] $) - b*det($ \left[\begin{array}{cccc}e&g&h\\i&k&l\\m&o&p\end{array}\right] $) + c*det($ \left[\begin{array}{cccc}e&f&h\\i&j&l\\m&n&p\end{array}\right] $)

Finally, subtract this by the number in the first row and fourth column multiplied by the determinant of the 3 x 3 matrix created when you omit the first row and fourth column.

 a*det($ \left[\begin{array}{cccc}f&g&h\\j&k&l\\n&o&p\end{array}\right] $) - b*det($ \left[\begin{array}{cccc}e&g&h\\i&k&l\\m&o&p\end{array}\right] $) + c*det($ \left[\begin{array}{cccc}e&f&h\\i&j&l\\m&n&p\end{array}\right] $) - d*det($ \left[\begin{array}{cccc}e&f&g\\i&j&k\\m&n&o\end{array}\right] $)

Therefore...

det(A) = a*[f*(k*p - l*o) - g*(j*p - l*n) + h*(j*o - k*n)] - b*[e*(k*p - l*o) - g*(i*p - l*m) + h*(i*o - k*m)] + c*[e*(j*p - l*n) - f*(i*p - l*m) + h*(i*n- j*m)] - d*[e*(j*o - k*n) - f*(i*o - k*m) + g*(i*n - j*m)]




Examples

2 x 2 Matrix

A = $ \left[\begin{array}{cccc}4&2\\-3&1\end{array}\right] $

det(A) = (4)*(1) - (2)*(-3)

      = 4-(-6)
      = 10

3 x 3 Matrix

A = $ \left[\begin{array}{cccc}2&5&1\\3&2&2\\1&5&2\end{array}\right] $

det(A) = 2*[(2*2) - (2*5)] - 5*[(3*2) - (2*1)] + 1*[(3*5) - (2*1)]

      = 2*(4 - 10) - 5*(6 - 2) + 1*(15 - 2)
      = 2*(-6) - 5*(4) + 1*(13)
      = -12 - 20 + 13
      = -19

4 x 4 Matrix

A = $ \left[\begin{array}{cccc}2&3&7&2\\1&6&2&3\\1&1&4&3\\4&6&5&1\end{array}\right] $

det(A) = 2*{6*[(4*1) - (3*5)] - 2*[(1*1) - (3*6)] + 3*[(1*5) - (4*6)]} - 3*{1*[(4*1) - (3*5)] - 2*[(1*1) - (3*4)] + 3*[(1*5) - (4*4)]} + 7*{1*[(1*1) - (3*6)] - 6*[(1*1) - (3*4)] + 3*[(1*6) - (1*4)]} - 2*{1*[(1*5) - (4*6)] - 6*[(1*5) - (4*4)] + 2*[(1*6) - (1*4)]}

      = 2*[6*(4-15) - 2*(1-18) + 3*(5-24)] - 3*[1*(4-15) - 2*(1-12) + 3*(5-16)] + 7*[1*(1-18) - 6*(1-12) + 3*(6-4)] -
        2*[1*(5-24) - 6*(5-16) + 2*(6-4)]
      = 2*[(6*(-11)) - (2*(-17)) + (3*(-19))] - 3*[(1*(-11)) - (2*(-11)) + (3*(-11))] + 7*[(1*(-17)) - (6*(-11)) + (3*2)] -
        2*[(1*(-19)) - (6*(-11)) + (2*2)]
      = 2*(-66 - (-34) + (-57)) - 3*(-11 - (-22) + (-33)) + 7*((-17) - (-66) + 6) - 2*((-19) - (-66) + 4)
      = 2*(-89) - 3*(-22) + 7*(55) - 2*(51)
      = -178 - (-66) + 385 - 102
      = 171

Important Properties of the Determinant

- The determinant of the transpose of a matrix will be the same as the determinant of the original matrix.

det(A^T) = det(A)

- If two rows in the matrix are interchanged, the determinant will be the negative of the determinant of the original matrix.

Eg. If A = $ \left[\begin{array}{cccc}2&5&1\\3&2&2\\1&5&2\end{array}\right] $ and B = $ \left[\begin{array}{cccc}2&5&1\\1&5&2\\3&2&2\end{array}\right] $ , then det(A) = -det(B) = 19

- If two rows or columns in a matrix are the same, the determinant will be zero.

Eg. If A = $ \left[\begin{array}{cccc}2&5&2\\3&2&2\\1&5&2\end{array}\right] $ , then det(A) = 0

- If there is a row of zeros in the matrix, the determinant will be zero.

Eg. If A = $ \left[\begin{array}{cccc}2&3&1\\3&3&2\\1&3&2\end{array}\right] $ , then det(A) = 0

- If a row or column in a matrix was multiplied by a real number, then the determinant will be that real number multiplied by the determinant of the original matrix.

Eg. If kA = $ \left[\begin{array}{cccc}2&5&k1\\3&2&k2\\1&5&k2\end{array}\right] $ , then det(A) = k(det(A)) = -19

- If the matrix is either an upper or a lower triangular matrix, the determinant will be the product of the elements on the main diagonal.

Eg. If A = $ \left[\begin{array}{cccc}2&0&0\\0&2&0\\0&0&2\end{array}\right] $ , then det(A) = 2*2*2 = 8

- If two matrices are n x n, then the determinant of one multiplied by the determinant of the other is the determinant of the full matrix of the two next to each other.

Eg. If A = $ \left[\begin{array}{cccc}2&5&1\\3&2&2\\1&5&2\end{array}\right] $ and B = $ \left[\begin{array}{cccc}1&2&8\\4&0&4\\6&1&2\end{array}\right] $ , then det(A)*det(B) = det(AB) = 1140

NOTE: further examples of these properties and more ways they can be used can be found on another Rhea page [[1]]


Nonsingular and Similar Matrices

Definition: A square matrix is called nonsingular if its determinant is not equal to zero.

- For square matrices, then the linear combination of the matrix has a nontrivial solution if and only if the matrix is NOT nonsingular.

- For nonsingular matrices, the determinant of the inverse of that matrix is equal to 1 over the determinant of the matrix.

- If two matrices are similar matrices, then the determinants of the two are equal.


For other applications and uses for the determinant, look here: [[2]]


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