document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164620>Question 1164620</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Let A and B be m × n matrices and let k be a scalar. Show that (10 points)<BR>\n" );
document.write( "(A^T + k B^T<BR>\n" );
document.write( ")(A + k B)<BR>\n" );
document.write( "is symmetric. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854338 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854338\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To show that the matrix $M = (A^T + k B^T)(A + k B)$ is symmetric, we must prove that the transpose of the entire expression is equal to the expression itself. That is, we must show that $M^T = M$.\r<BR>\n" );
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document.write( "### **Definitions and Properties**<BR>\n" );
document.write( "We will use the following fundamental properties of matrix transposes:<BR>\n" );
document.write( "1.  **Sum/Difference Rule:** $(X + Y)^T = X^T + Y^T$<BR>\n" );
document.write( "2.  **Product Rule:** $(XY)^T = Y^T X^T$<BR>\n" );
document.write( "3.  **Scalar Rule:** $(kX)^T = k(X^T)$<BR>\n" );
document.write( "4.  **Double Transpose Rule:** $(X^T)^T = X$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Proof**\r<BR>\n" );
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document.write( "Let $M = (A^T + k B^T)(A + k B)$.\r<BR>\n" );
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document.write( "**Step 1: Apply the transpose to the entire product.**<BR>\n" );
document.write( "Using the **Product Rule** $(XY)^T = Y^T X^T$, we swap the order of the two main factors and transpose them:<BR>\n" );
document.write( "$$M^T = \left[ (A^T + k B^T) (A + k B) \right]^T = (A + k B)^T (A^T + k B^T)^T$$\r<BR>\n" );
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document.write( "**Step 2: Distribute the transpose into the sums.**<BR>\n" );
document.write( "Using the **Sum Rule** and the **Scalar Rule** on both factors:<BR>\n" );
document.write( "$$(A + k B)^T = A^T + (kB)^T = A^T + k B^T$$<BR>\n" );
document.write( "$$(A^T + k B^T)^T = (A^T)^T + (kB^T)^T = A + k(B^T)^T = A + kB$$\r<BR>\n" );
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document.write( "**Step 3: Substitute the results back into the expression.**<BR>\n" );
document.write( "Now substitute the simplified factors back into the equation from Step 1:<BR>\n" );
document.write( "$$M^T = (A^T + k B^T) (A + k B)$$\r<BR>\n" );
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document.write( "**Step 4: Conclusion.**<BR>\n" );
document.write( "We observe that:<BR>\n" );
document.write( "$$M^T = (A^T + k B^T) (A + k B) = M$$\r<BR>\n" );
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document.write( "Since the transpose of the matrix is equal to the original matrix, the expression $(A^T + k B^T)(A + k B)$ is **symmetric**. $\blacksquare$ </SPAN>\n" );
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