CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Byju's Answer
Standard XII
Mathematics
Adjoint of a Matrix
Express the f...
Question

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

⎡⎢⎣33−1−2−21−4−52⎤⎥⎦

Open in App
Solution

Let A=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦, then A′=⎡⎢⎣3−2−43−2−5−112⎤⎥⎦
Now, A+A′=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦+⎡⎢⎣3−2−43−2−5−112⎤⎥⎦=⎡⎢⎣61−51−4−4−5−44⎤⎥⎦
P=12(A+A′)=12⎡⎢⎣61−51−4−4−5−44⎤⎥⎦=⎡⎢ ⎢ ⎢⎣312−−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦
Now, P′=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦=P
Thus, P=12(A+A′) is a symmetric matrix.
Now, A−A′=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦−⎡⎢⎣3−2−43−2−5−112⎤⎥⎦=⎡⎢⎣053−506−3−60⎤⎥⎦
Let Q=12(A−A′)=12⎡⎢⎣053−506−3−60⎤⎥⎦=⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦

Now, Q′=⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣0−52−32520−33230⎤⎥ ⎥ ⎥⎦=−Q
Thus, Q=12(A−A′) is a skew -symmetric matrix.
Representing A as the sum of P and Q.
P+Q=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦+⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦=A


flag
Suggest Corrections
thumbs-up
42
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Adjoint and Inverse of a Matrix
MATHEMATICS
Watch in App
explore_more_icon
Explore more
Adjoint of a Matrix
Standard XII Mathematics
Join BYJU'S Learning Program
book_image
Solve
book_image
Textbooks
book_image
Question Papers
book_image
Install app
CrossIcon