intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.
35. (a) If P = 120−131042 and Q = 2−4222−1−4−45, find (QP) and hence solve the following system of equations using matrices:
x−y=3, 2x+3y+4z=17, y+2z=7
OR
(b) Obtain the value of Δ=1+x1111+y1111+z in terms of x, y and z.
Further, if Δ=0 and x, y, z are non-zero real numbers, prove that x−1+y−1+z−1=−1.
(matrix, determinant, system of equations, inverse matrix, linear equations)
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Step-by-Step Solution
Step 1: Define the Determinant
We are given a 3×3 determinant and asked to find its value in terms of x, y, and z. This is the first step in solving the problem.
Step 2: Apply Column Operations to Simplify
To simplify the calculation of the determinant, we apply column operations. Subtracting the third column from the first and second columns will introduce zeros, making the expansion easier.
Step 3: Evaluate the Determinant after Column Operations
After applying the column operations, the determinant transforms into this new form. Notice that the first two columns now contain some zeros, which will simplify the expansion process.
Step 4: Expand the Determinant along the First Row
We expand the determinant along the first row. This involves multiplying each element in the first row by its corresponding cofactor. The terms with zero elements simplify the expansion by eliminating those cofactors immediately.
Step 5: Calculate the 2x2 Determinants
Now we calculate the 2×2 determinants resulting from the expansion. This involves the cross-multiplication of elements in each minor matrix and then combining them with the original row elements.
Step 6: Simplify and Express in Terms of x, y, z
We simplify the expression by performing the multiplications and combining like terms. This gives us the value of the determinant in terms of x, y, and z.
Step 7: Consider the Condition Δ=0
The problem states that if Δ=0, we need to prove a certain relationship. We set our derived expression for Δ equal to zero and proceed to the next step.
Step 8: Divide by xyz
Since x, y, z are non-zero real numbers, we can divide the entire equation by xyz. This operation is crucial for transforming the expression into the desired form.
Step 9: Simplify the Expression
After dividing each term by xyz, we simplify the fractions. This results in an expression containing the reciprocals of x, y, and z, along with the constant 1.
Step 10: Rearrange to Prove the Identity
Finally, we rearrange the terms to isolate the sum of reciprocals on one side of the equation. This proves the required identity, demonstrating that x−1+y−1+z−1=−1.