Two by Two

Algebra Level 3

Consider a 2 by 2 matrix C C given by [ a c b d ] \begin{bmatrix} a & c \\ b & d \end{bmatrix} where a a , b b , c c and d d are real numbers.

If C C has the property that its inverse is equal to its transpose, i.e. C − 1 = C T C^{-1}=C^{T} , then what is the value of a 2 + b 2 + c 2 + d 2 a^2+b^2+c^2+d^2 ?


The answer is 2.

Jaydee Lucero by Jaydee Lucero , 24, Philippines

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1 solution

Noé Zelaya
May 28, 2015

From:

A − 1 = A T A^{-1} = A^{T}

Multiply both sides by A

A . A − 1 = A . A T A . A^{-1} = A . A^{T}

I = A . A T I = A . A^{T}

Where I is the grade 2 identity matrix. Operating A . A T A . A^{T} : ( a 2 + c 2 a . c + b . d b . c + a . d b 2 + d 2 ) \begin{pmatrix} a^{2}+c^{2}&a.c+b.d \\b.c+a.d & b^{2}+d^{2}\end{pmatrix} = ( 1 0 0 1 ) \begin{pmatrix} 1&0 \\0&1 \end{pmatrix}

Then:

a 2 + c 2 = 1 a^{2}+c^{2}=1 and b 2 + d 2 = 1 b^{2}+d^{2}=1 .

Therefore a 2 + b 2 + c 2 + d 2 = 1 a^{2}+b^{2}+c^{2}+d^{2}=1 .

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