Multiply the Matrices

Problem

[[−1/296,9/148,0],[17/296,−5/148,0],[−1/53,−2/6095,296/6095]]*[[21],[24],[14]]

Solution

1. Set up the matrix multiplication by multiplying each row of the first matrix by the column vector.

2. Calculate the first element of the resulting vector.

−1/296*(21)+9/148*(24)+0*(14)

−21/296+216/148

−21/296+432/296=411/296

3. Calculate the second element of the resulting vector.

17/296*(21)−5/148*(24)+0*(14)

357/296−120/148

357/296−240/296=117/296

4. Calculate the third element of the resulting vector.

−1/53*(21)−2/6095*(24)+296/6095*(14)

−21/53−48/6095+4144/6095

−2415/6095−48/6095+4144/6095=1681/6095

5. Simplify the fractions where possible.

1681/6095=(41⋅41)/(41⋅148.658...)* (No common integer factors for 411/296 or 117/296)

1681/6095=(31⋅54.2...)/(5⋅1219)⇒1681/6095=41/(148.6...)* (Check 6095/41 = 148.65)

1681/6095=(58⋅29)/(5⋅1219)⇒1681/6095=(58⋅29)/(5⋅23⋅53)

1681/6095=41/(5⋅23⋅53)

Final Answer

[[−1/296,9/148,0],[17/296,−5/148,0],[−1/53,−2/6095,296/6095]]*[[21],[24],[14]]=[[411/296],[117/296],[1681/6095]]

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