Properties of Determinant of Matrix Class 12

Here you will learn properties of determinant of matrix with examples.

Let’s begin –

Properties of Determinant of Matrix

Property 1 :

The value of determinant remains unaltered or unchanged, if the rows & columns are inter-changed,

e.g. if D = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) = \(\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}\)

Property 2 :

If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.

Let D = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) & \(D_1\) = \(\begin{bmatrix}  a_2 & b_2 & c_2 \\ a_1 & b_1 & c_1  \\ a_3 & b_3 & c_3 \end{bmatrix}\). Then \(D_1\) = -D.

Property 3 :

If all the elements of a row (or column) are zero, then the value of the determinant is zero.

Property 4 :

If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.

e.g.  If D = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) and \(D_1\) = \(\begin{bmatrix} ka_1 & kb_1 & kc_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\). Then \(D_1\) = k D.

Property 5 :

If all the elements of a row (or column) are proportional (or identical) to the elements of any other row, then the determinant vanishes, i.e. its value is zero.

e.g. If D = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) \(\implies\) D = 0

If \(D_1\)=- \(\begin{bmatrix} a_1 & b_1 & c_1 \\ ka_1 & kb_1 & kc_1 \\ a_3 & b_3 & c_3 \end{bmatrix}\). \(\implies\) \(D_1\) = 0.

Property 6 :

If each element of any row (or column) is expressed as sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.

e.g. \(\begin{bmatrix} a_1 + x & b_1 + y & c_1 + z \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) + \(\begin{bmatrix} x & y & z \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\)

Property 7 :

Row – column Operation : The value of a determinant remains unaltered under a column (\(C_i\)) operation of the form \(C_i\) \(\rightarrow\) \(C_i\) + \(\alpha C_j\)  (j \(\ne\) i) or row (\(R_i\)) operation of the form \(R_i\) \(\rightarrow\) \(R_i\) + \(\alpha R_j\)  (j \(\ne\) i). In other words, the value of determinant  is not altered by adding elements of any row ( or column) to the same multiples of the corresponding elements of any other row (or column).

Property 8 :

If the elements of a determinant D are rational integral functions of x and two rows (or columns) become identical when x = a then (x – a) is a factor of D.

Note that if rows become identical when a is substituted for x, then \((x-a)^{r-1}\) is a factor of D.

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