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   <h2 class="likechapterHead"><a 
 id="x29-102000"></a>Chapter M&#x00A0;&#x00A0;Matrices</h2>
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</p><!--l. 10--><p class="indent" >   We have made frequent use of matrices for solving systems of equations, and
we have begun to investigate a few of their properties, such as the null space and
nonsingularity. In this chapter, we will take a more systematic approach to the
study of matrices.
</p>
   <div class="likesectionTOCS">
   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-103000">Section MO Matrix Operations</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-104000" id="QQ2-30-104">Subsection MEASM: Matrix Equality, Addition, Scalar Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-105000" id="QQ2-30-105">Subsection VSP: Vector Space Properties</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-106000" id="QQ2-30-106">Subsection TSM: Transposes and Symmetric Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-107000" id="QQ2-30-107">Subsection MCC: Matrices and Complex Conjugation</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-108000" id="QQ2-30-108">Subsection AM: Adjoint of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-109000" id="QQ2-30-109">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-110000" id="QQ2-30-110">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li29.xml#x30-111000" id="QQ2-30-111">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-112000">Section MM Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-113000" id="QQ2-31-113">Subsection MVP: Matrix-Vector Product</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-114000" id="QQ2-31-114">Subsection MM: Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-115000" id="QQ2-31-115">Subsection MMEE: Matrix Multiplication, Entry-by-Entry</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-116000" id="QQ2-31-116">Subsection PMM: Properties of Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-117000" id="QQ2-31-117">Subsection HM: Hermitian Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-118000" id="QQ2-31-118">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-119000" id="QQ2-31-119">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li30.xml#x31-120000" id="QQ2-31-120">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-121000">Section MISLE Matrix Inverses and Systems of Linear Equations</a></span>
                                                                          

                                                                          
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-122000" id="QQ2-32-122">Subsection IM: Inverse of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-123000" id="QQ2-32-123">Subsection CIM: Computing the Inverse of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-124000" id="QQ2-32-124">Subsection PMI: Properties of Matrix Inverses</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-125000" id="QQ2-32-125">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-126000" id="QQ2-32-126">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li31.xml#x32-127000" id="QQ2-32-127">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-128000">Section MINM Matrix Inverses and Nonsingular Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-129000" id="QQ2-33-129">Subsection NMI: Nonsingular Matrices are Invertible</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-130000" id="QQ2-33-130">Subsection UM: Unitary Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-131000" id="QQ2-33-131">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-132000" id="QQ2-33-132">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li32.xml#x33-133000" id="QQ2-33-133">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-134000">Section CRS Column and Row Spaces</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-135000" id="QQ2-34-135">Subsection CSSE: Column Spaces and Systems of Equations</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-136000" id="QQ2-34-136">Subsection CSSOC: Column Space Spanned by Original Columns</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-137000" id="QQ2-34-137">Subsection CSNM: Column Space of a Nonsingular Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-138000" id="QQ2-34-138">Subsection RSM: Row Space of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-139000" id="QQ2-34-139">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-140000" id="QQ2-34-140">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li33.xml#x34-141000" id="QQ2-34-141">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-142000">Section FS Four Subsets</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-143000" id="QQ2-35-143">Subsection LNS: Left Null Space</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-144000" id="QQ2-35-144">Subsection CRS: Computing Column Spaces</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-145000" id="QQ2-35-145">Subsection EEF: Extended echelon form</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-146000" id="QQ2-35-146">Subsection FS: Four Subsets</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-147000" id="QQ2-35-147">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-148000" id="QQ2-35-148">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-149000" id="QQ2-35-149">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-1.32li34.xml#x35-150000" id="QQ2-35-150">Annotated Acronyms M: Matrices</a></span>
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