Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R , Suppose Q = q i j is a matrix such that P Q   =   k I , where k ∈ R ,   k ≠ 0 and I is the identity matrix of order 3. If q 23 =   - k 8 and det ⁡ Q = k 2 2 then

Question

Let P = 3 - 1 - 2 2 0 &#945; 3 - 5 0 , where &#945; &#8712; R , Suppose Q = q i j is a matrix such that P Q &#160; = &#160; k I , where k &#8712; R , &#160; k &#8800; 0 and I is the identity matrix of order 3. If q 23 = &#160; - k 8 and det &#8289; Q = k 2 2 then

Aicharya · Accepted Answer

P Q = k I P Q = k 3&#160; &#160; &#160; ( as order is 3 ) &#8658; P = 2 k &#160; &#8800; 0 &#160; &#160; &#8658; &#160; &#160; P is an invertible matrix &#8757; P Q = k I &#8756; Q = k P - 1 &#160; I = K 1 P adj P &#8756; Q = a d j . P 2 &#8757; q 23 = &#160; - k 8 ( &#8756; cofactor P 32 = - 3 &#945; + 4 ) &#8658; 12 &#945; + 16 = K &#38; adjP 2 = P 32 2 T - q 23 &#8756; undefined ...... (i) Also P = 2 k &#8658; k = 10 + 6 &#945; ......(ii) Solving these two &#8658; 12 &#945; + 16 = 10 + 6 &#945; &#8658; &#945; = - 1 & K = 4 &#8756; 4 &#945; - k + 8 = 0 ( &#8658; 4 ( &#8722; 1 ) &#8722; ( 4 ) + 8 = 0 &#8658; &#8722; 4 &#8722; 4 + 8 = 0 &#8658; &#8722; 8 + 8 = 0 ) And det &#8289; P a d j (Q = P &#160; a d j (Q) = 2 k . &#160; k 2 2 2 = k 5 2 = 2 9 Also det Q adj P = Q adj P = k 2 2 2k 2 = 2k 4 = 2 9

Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R , Suppose Q = q i j is a matrix such that P Q…

Solution

Let P = 3 - 1 - 2 2 0 &#945; 3 - 5 0 , where &#945; &#8712; R , Suppose Q = q i j is a matrix such that P Q…

Solution

Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R , Suppose Q = q i j is a matrix such that P Q…