Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R . Suppose Q = q i j is a matrix satisfying P Q = k I 3 for some non-zero k ∈ R . If q 23 = - k 8 and Q = k 2 2 , then α 2 + k 2 is equal to_________.

Question

Let P = 3 - 1 - 2 2 0 &#945; 3 - 5 0 , where &#945; &#8712; R . Suppose Q = q i j is a matrix satisfying P Q = k I 3 for some non-zero k &#8712; R . If q 23 = - k 8 and Q = k 2 2 , then &#945; 2 + k 2 is equal to_________.

Quantrex Academy · Accepted Answer

Correct answer: 0

P Q = k I P . Q = k 3 &#8658; &#160; P = 2 k &#8800; 0 &#8658; P is an invertible matrix &#8757; &#160; P Q = k I &#8756; &#160; Q = k P - 1 I &#8756; &#160; Q = &#160; adj . P 2 &#8757; &#160; q 23 = - k 8 &#8756; &#160; - 3 &#945; + 4 2 = - k 8 &#8658; k = 4 &#8756; &#160; P = 2 k &#8658; k = 10 + 6 &#945; &#160; &#160; &#160; &#160; &#8230; i Put value of k in i .. we get &#945; = - 1 &#8658; &#945; 2 + k 2 = 17

Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R . Suppose Q = q i j is a matrix satisfying P Q = k I 3 for some non-zero k ∈ R . If q 23 = - k 8 and Q = k 2 2 , then α 2 + k 2 is equal to_________.

Correct answer

Step-by-step solution

More from Matrices

Let P = 3 - 1 - 2 2 0 &#945; 3 - 5 0 , where &#945; &#8712; R . Suppose Q = q i j is a matrix satisfying P Q = k I 3 for some non-zero k &#8712; R . If q 23 = - k 8 and Q = k 2 2 , then &#945; 2 + k 2 is equal to_________.

Correct answer

Step-by-step solution

More from Matrices

Let P = 3 - 1 - 2 2 0 α 3 - 5 0 , where α ∈ R . Suppose Q = q i j is a matrix satisfying P Q = k I 3 for some non-zero k ∈ R . If q 23 = - k 8 and Q = k 2 2 , then α 2 + k 2 is equal to_________.