Show that A+A^T is symmetric for any 2 x 2 matrix A. Is this true for any square matrix?<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Correct Answer:<\/strong><\/p>\n\n\n\n Yes, for any square matrix AA, the matrix A+ATA + A^T is symmetric.<\/p>\n\n\n\n To determine whether A+ATA + A^T is symmetric, we must recall the definition of a symmetric matrix<\/strong>:<\/p>\n\n\n\n A matrix MM is symmetric if MT=MM^T = M, where MTM^T is the transpose of MM.<\/p>\n<\/blockquote>\n\n\n\n Let\u2019s test this with a general 2 \u00d7 2 matrix<\/strong>: A=[abcd]A = \\begin{bmatrix} a & b \\\\ c & d \\\\ \\end{bmatrix}<\/p>\n\n\n\n Then, the transpose<\/strong> of AA is: AT=[acbd]A^T = \\begin{bmatrix} a & c \\\\ b & d \\\\ \\end{bmatrix}<\/p>\n\n\n\n Now, compute A+ATA + A^T: A+AT=[abcd]+[acbd]=[a+ab+cc+bd+d]=[2ab+cb+c2d]A + A^T = \\begin{bmatrix} a & b \\\\ c & d \\\\ \\end{bmatrix} + \\begin{bmatrix} a & c \\\\ b & d \\\\ \\end{bmatrix} = \\begin{bmatrix} a + a & b + c \\\\ c + b & d + d \\\\ \\end{bmatrix} = \\begin{bmatrix} 2a & b + c \\\\ b + c & 2d \\\\ \\end{bmatrix}<\/p>\n\n\n\n This matrix is symmetric because the transpose of A+ATA + A^T is: (A+AT)T=[2ab+cb+c2d]T=[2ab+cb+c2d]=A+AT(A + A^T)^T = \\begin{bmatrix} 2a & b + c \\\\ b + c & 2d \\\\ \\end{bmatrix}^T = \\begin{bmatrix} 2a & b + c \\\\ b + c & 2d \\\\ \\end{bmatrix} = A + A^T<\/p>\n\n\n\n Therefore, A+ATA + A^T is symmetric.<\/p>\n\n\n\n The above result isn’t limited to 2 \u00d7 2 matrices. For any square matrix AA<\/strong> (i.e., of size n\u00d7nn \\times n), we use the fact that: (A+AT)T=AT+(AT)T=AT+A=A+AT(A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T<\/p>\n\n\n\n Since the transpose of A+ATA + A^T equals itself, it is symmetric<\/strong>.<\/p>\n\n\n\n Yes<\/strong>, A+ATA + A^T is symmetric for any square matrix AA<\/strong>, including 2 \u00d7 2 matrices.<\/p>\n","protected":false},"excerpt":{"rendered":" Show that A+A^T is symmetric for any 2 x 2 matrix A. Is this true for any square matrix? The correct answer and explanation is: Correct Answer: Yes, for any square matrix AA, the matrix A+ATA + A^T is symmetric. Explanation (300 words): To determine whether A+ATA + A^T is symmetric, we must recall the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-24623","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24623","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=24623"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24623\/revisions"}],"predecessor-version":[{"id":24628,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24623\/revisions\/24628"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=24623"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=24623"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=24623"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nExplanation (300 words):<\/strong><\/h3>\n\n\n\n
\n
General Case: Any Square Matrix<\/strong><\/h3>\n\n\n\n
Conclusion:<\/strong><\/h3>\n\n\n\n