Linear Algebra

Section IV. Matrix Operations 217After zero matrices, the matrices whose actions are easiest to understandare the ones with a single nonzero entry.3.2 Definition A matrix with all zeroes except for a one in the i, j entry isan i, j unit matrix.3.3 Example This is the 1, 2 unit matrix with three rows and two columns,multiplying from the left.⎛⎝ 0 1⎞ ⎛( )0 0⎠5 6= ⎝ 7 8⎞0 0⎠7 80 00 0Acting from the left, an i, j unit matrix copies row j of the multiplicand intorow i of the result. From the right an i, j unit matrix copies column i of themultiplicand into column j of the result.⎛⎝ 1 2 3⎞ ⎛4 5 6⎠⎝ 0 1⎞ ⎛0 0⎠ = ⎝ 0 1⎞0 4⎠7 8 9 0 0 0 73.4 Example Rescaling these matrices simply rescales the result. This is theaction from the left of the matrix that is twice the one in the prior example.⎛⎝ 0 2⎞ ⎛ ⎞( ) 14 160 0⎠ 5 6= ⎝ 0 0 ⎠7 80 00 0And this is the action of the matrix that is minus three times the one from theprior example. ⎛⎝ 1 2 3⎞ ⎛4 5 6⎠⎝ 0 −3⎞ ⎛0 0 ⎠ = ⎝ 0 −3⎞0 −12⎠7 8 9 0 0 0 −21Next in complication are matrices with two nonzero entries. There are twocases. If a left-multiplier has entries in different rows then their actions don’tinteract.3.5 Example⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ 1 0 00 0 2⎠⎝ 1 2 34 5 6⎠ = ( ⎝ 1 0 00 0 0⎠ + ⎝ 0 0 00 0 2⎠)⎝ 1 2 34 5 6⎠0 0 0 7 8 9 0 0 0 0 0 0 7 8 9⎛ ⎞ ⎛ ⎞1 2 3 0 0 0= ⎝0 0 0⎠ + ⎝14 16 18⎠0⎛0 0⎞0 0 0= ⎝ 1 2 314 16 18⎠0 0 0