To find the product of the matrices [tex]\( X = \begin{pmatrix} -2 & 2 \\ 1 & -3 \end{pmatrix} \)[/tex] and [tex]\( Y = \begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix} \)[/tex], we'll perform matrix multiplication by taking the dot product of the rows of [tex]\( X \)[/tex] with the columns of [tex]\( Y \)[/tex].
Let's find each entry in the resulting matrix [tex]\( XY \)[/tex] step-by-step.
### Entry [tex]\( a \)[/tex]:
To find [tex]\( a \)[/tex], we take the dot product of the first row of [tex]\( X \)[/tex] with the first column of [tex]\( Y \)[/tex]:
[tex]\[ a = (-2 \cdot 4) + (2 \cdot -3) \][/tex]
[tex]\[ a = -8 + (-6) \][/tex]
[tex]\[ a = -14 \][/tex]
### Entry [tex]\( b \)[/tex]:
To find [tex]\( b \)[/tex], we take the dot product of the first row of [tex]\( X \)[/tex] with the second column of [tex]\( Y \)[/tex]:
[tex]\[ b = (-2 \cdot -1) + (2 \cdot 2) \][/tex]
[tex]\[ b = 2 + 4 \][/tex]
[tex]\[ b = 6 \][/tex]
### Entry [tex]\( c \)[/tex]:
To find [tex]\( c \)[/tex], we take the dot product of the second row of [tex]\( X \)[/tex] with the first column of [tex]\( Y \)[/tex]:
[tex]\[ c = (1 \cdot 4) + (-3 \cdot -3) \][/tex]
[tex]\[ c = 4 + 9 \][/tex]
[tex]\[ c = 13 \][/tex]
### Entry [tex]\( d \)[/tex]:
To find [tex]\( d \)[/tex], we take the dot product of the second row of [tex]\( X \)[/tex] with the second column of [tex]\( Y \)[/tex]:
[tex]\[ d = (1 \cdot -1) + (-3 \cdot 2) \][/tex]
[tex]\[ d = -1 + (-6) \][/tex]
[tex]\[ d = -7 \][/tex]
Putting it all together, the product [tex]\( XY \)[/tex] is:
[tex]\[ XY = \begin{pmatrix} -14 & 6 \\ 13 & -7 \end{pmatrix} \][/tex]
Thus, the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are:
[tex]\( a = -14 \)[/tex], [tex]\( b = 6 \)[/tex], [tex]\( c = 13 \)[/tex], [tex]\( d = -7 \)[/tex].
