Exams Joint Entrance Examination Mathematics Matrices and Determinants Question 107 Question 107 of 472 Medium NUMERICAL
Consider the matrices A = [ 2 − 2 4 − 2 ] A=[\begin{array}{|ccc|}2 & -2 \\ 4 & -2\end{array}] A = [ 2 4 − 2 − 2 ] and B = [ 3 9 1 3 ] B=[\begin{array}{|ccc|}3 & 9 \\ 1 & 3\end{array}] B = [ 3 1 9 3 ] . If matrices P P P and Q Q Q are such that P A = B PA=B P A = B and A Q = B AQ=B A Q = B , then the absolute value of the sum of the diagonal elements of 2 ( P + Q ) 2(P+Q) 2 ( P + Q ) is \_\_\_\_\_\_\_\_.
Answer: 34
Solutions · 1 Post your solution Official solution
{A}^{-1}=\frac{1}{4}[\begin{gathered} -2 2 \\ -4 2 ]=\frac{1}{2}[ -1 1 \\ -2 1 ]
∴ P A = B \therefore PA=B ∴ P A = B
⇒ P = B A − 1 \Rightarrow P=B{A}^{-1} ⇒ P = B A − 1
= 1 2 [ 39 13 ] [ − 11 − 21 ] = 1 2 [ − 2112 − 74 ] =\frac{1}{2}[ 3 9 \\ 1 3 ][ -1 1 \\ -2 1 ]=\frac{1}{2}[ -21 12 \\ -7 4 ] = 2 1 [ 39 13 ] [ − 11 − 21 ] = 2 1 [ − 2112 − 74 ]
∴ A Q = B \therefore AQ=B ∴ A Q = B
⇒ Q = A − 1 B \Rightarrow Q={A}^{-1}B ⇒ Q = A − 1 B
=\frac{1}{2}[ -1 1 \\ -2 1 ][ 3 9 \\ 1 3 ]=\frac{1}{2}[ -2 -6 \\ -5 -15 \end{gathered}]
∴ ∣ t r ( 2 ( P + Q ) ) ∣ = ∣ ( − 21 ) + ( 4 ) + ( − 2 ) + ( − 15 ) ∣ \therefore |tr(2(P+Q))|=|(-21)+(4)+(-2)+(-15)| ∴ ∣ t r ( 2 ( P + Q )) ∣ = ∣ ( − 21 ) + ( 4 ) + ( − 2 ) + ( − 15 ) ∣
= ∣ − 34 ∣ = 34 =|-34|=34 = ∣ − 34∣ = 34 Answer the question to unlock