Math 115 Midterm Review Problems1.(a) Solve the following linear system:x+y+2z=22x+y-z=3x+2y+7z=3(b) Find the solution to the associated homogeneous system:x+y+2z=02x+y-z=0x+2y+7z=02. SupposeA,BandCare invertiblen×nmatrices. Prove the following, using the definition of matrixinverse:(A-1BC)-1=C-1B-1A3. Find the values of the numbercsuch that1c020cc-11has an inverse.4. Find the inverse of1-1-2-101210,and use it to solve the systemx-y-2z=3-x+z=02x+y=15. Consider the transformationT:R2→R2defined as follows:Counterclockwise rotation about the origin through an angle ofπ/2 followed by reflection in the liney=x.Determine the standard matrix forTand findT12.6. IfTA:R2→R2is a linear transformation defined byTAxy=x+yx-yandTB:R2→R2is a lineartransformation defined byTBxy=3x2x+ 4y, find(a) the standard matrix forTA◦TB.
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