Question

Plsss solve this question of determinants i will give you 30+ points and Mark as brainleist​

Answer 1

$\underline{\text{Proof :}}$

$\text{L.H.S.}=\left | \begin{array}{ccc}a+b&b+c&c+a\\p+q&q+r&r+p\\l+m&m+n&n+l\end{array}\right |$

$\underline{\mathrm{C'_{1}\to (C_{1}+C_{2}+C_{3})\:gives}}$

$=\left | \begin{array}{ccc}2(a+b+c)&b+c&c+a\\2(p+q+r)&q+r&r+p\\2(l+m+n)&m+n&n+l\end{array}\right |$

$=2\left | \begin{array}{ccc}a+b+c&b+c&c+a\\p+q+r&q+r&r+p\\l+m+n&m+n&n+l\end{array}\right |$

$\underline{\mathrm{C'_{1}\to (C_{1}-C_{2})\:gives}}$

$\small{=2\left | \begin{array}{ccc}a+b+c-b-c&b+c&c+a\\p+q+r-p-q&q+r&r+p\\l+m+n-l-m&m+n&n+l\end{array}\right |}$

$=2\left | \begin{array}{ccc}a&b+c&c+a\\p&q+r&r+p\\l&m+n&n+l\end{array}\right |$

$\underline{\mathrm{C'_{3}\to (C_{3}-C_{1})\:gives}}$

$=2\left | \begin{array}{ccc}a&b+c&c+a-a\\p&q+r&r+p-p\\l&m+n&n+l-l\end{array}\right |$

$=2\left | \begin{array}{ccc}a&b+c&c\\p&q+r&r\\l&m+n&n\end{array}\right |$

$\underline{\mathrm{C'_{2}\to (C_{2}-C_{3})\:gives}}$

$=2\left | \begin{array}{ccc}a&b+c-c&c\\p&q+r-r&r\\l&m+n-n&n\end{array}\right |$

$=2\left | \begin{array}{ccc}a&b&c\\p&q&r\\l&m&n\end{array}\right |=\text{R.H.S.}$

$\text{Hence, proved.}$

Answer 2

$\huge\bf\pink{\mid{\overline{\underline{Your\: Answer}}}\mid}$

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♦ IT IS THE QUESTION BASED ON ROW AND COLUMNS TRANSFORMATION OF DETERMINANTS.

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♣What is $\bold{DETERMINANTS}$ ?

✍️ In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|.

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Now,

in the given question,

♣ $\bold{PROPERTIES}$ OF DETERMINANTS USED ARE :-

✍️ If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

✍️ If some or all elements of a row or column of a determinant are expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.

✍️ If the equimultiples of corresponding elements of other rows (or columns) are added to every element of any row or column of a determinant, then the value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation

Ri → Ri + k Rj

 or

Ci → Ci + k Cj .

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♣$\bold{SOLUTION}$ to the question :-



Step I. Transform $C_1->(C_1+C_2+C_3)$

Step II. Transform $C_1->(C_1-C_2)$

Step III. Transform $C_3->(C_3-C_1)$

Step IV. Transform $C_2->(C_2-C_3)$

Note:-

All the steps and solution is in the attachments.

Kindly refer to it.✍️✍️

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