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Sagot :
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for which the sum of matrices
[tex]\[ A = \begin{pmatrix} 2x - 1 & 5 \\ 2y + 1 & 2 \end{pmatrix} \][/tex]
and
[tex]\[ B = \begin{pmatrix} 2 & 2x - 5 \\ 3 & x - 1 \end{pmatrix} \][/tex]
will equal the identity matrix
[tex]\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \][/tex]
we need to set up and solve the equation [tex]\( A + B = I \)[/tex].
First, calculate the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{pmatrix} 2x - 1 & 5 \\ 2y + 1 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 2x - 5 \\ 3 & x - 1 \end{pmatrix} = \begin{pmatrix} (2x - 1) + 2 & 5 + (2x - 5) \\ (2y + 1) + 3 & 2 + (x - 1) \end{pmatrix}. \][/tex]
Simplify each element in the resulting matrix:
[tex]\[ A + B = \begin{pmatrix} 2x - 1 + 2 & 5 + 2x - 5 \\ 2y + 1 + 3 & 2 + x - 1 \end{pmatrix} = \begin{pmatrix} 2x + 1 & 2x \\ 2y + 4 & x + 1 \end{pmatrix}. \][/tex]
We require this sum to equal the identity matrix [tex]\( I \)[/tex], therefore:
[tex]\[ \begin{pmatrix} 2x + 1 & 2x \\ 2y + 4 & x + 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \][/tex]
Now, equate the corresponding elements of the matrices:
1. [tex]\( 2x + 1 = 1 \)[/tex]
2. [tex]\( 2x = 0 \)[/tex]
3. [tex]\( 2y + 4 = 0 \)[/tex]
4. [tex]\( x + 1 = 1 \)[/tex]
Solve these equations step by step:
1. From [tex]\( 2x + 1 = 1 \)[/tex]:
[tex]\[ 2x + 1 = 1 \implies 2x = 0 \implies x = 0. \][/tex]
2. From [tex]\( 2x = 0 \)[/tex]:
[tex]\[ 2x = 0 \implies x = 0. \][/tex]
3. From [tex]\( 2y + 4 = 0 \)[/tex]:
[tex]\[ 2y + 4 = 0 \implies 2y = -4 \implies y = -2. \][/tex]
4. From [tex]\( x + 1 = 1 \)[/tex]:
[tex]\[ x + 1 = 1 \implies x = 0. \][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that will make the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] equal to the identity matrix are [tex]\( x = 0 \)[/tex] and [tex]\( y = -2 \)[/tex].
[tex]\[ A = \begin{pmatrix} 2x - 1 & 5 \\ 2y + 1 & 2 \end{pmatrix} \][/tex]
and
[tex]\[ B = \begin{pmatrix} 2 & 2x - 5 \\ 3 & x - 1 \end{pmatrix} \][/tex]
will equal the identity matrix
[tex]\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \][/tex]
we need to set up and solve the equation [tex]\( A + B = I \)[/tex].
First, calculate the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{pmatrix} 2x - 1 & 5 \\ 2y + 1 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 2x - 5 \\ 3 & x - 1 \end{pmatrix} = \begin{pmatrix} (2x - 1) + 2 & 5 + (2x - 5) \\ (2y + 1) + 3 & 2 + (x - 1) \end{pmatrix}. \][/tex]
Simplify each element in the resulting matrix:
[tex]\[ A + B = \begin{pmatrix} 2x - 1 + 2 & 5 + 2x - 5 \\ 2y + 1 + 3 & 2 + x - 1 \end{pmatrix} = \begin{pmatrix} 2x + 1 & 2x \\ 2y + 4 & x + 1 \end{pmatrix}. \][/tex]
We require this sum to equal the identity matrix [tex]\( I \)[/tex], therefore:
[tex]\[ \begin{pmatrix} 2x + 1 & 2x \\ 2y + 4 & x + 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \][/tex]
Now, equate the corresponding elements of the matrices:
1. [tex]\( 2x + 1 = 1 \)[/tex]
2. [tex]\( 2x = 0 \)[/tex]
3. [tex]\( 2y + 4 = 0 \)[/tex]
4. [tex]\( x + 1 = 1 \)[/tex]
Solve these equations step by step:
1. From [tex]\( 2x + 1 = 1 \)[/tex]:
[tex]\[ 2x + 1 = 1 \implies 2x = 0 \implies x = 0. \][/tex]
2. From [tex]\( 2x = 0 \)[/tex]:
[tex]\[ 2x = 0 \implies x = 0. \][/tex]
3. From [tex]\( 2y + 4 = 0 \)[/tex]:
[tex]\[ 2y + 4 = 0 \implies 2y = -4 \implies y = -2. \][/tex]
4. From [tex]\( x + 1 = 1 \)[/tex]:
[tex]\[ x + 1 = 1 \implies x = 0. \][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that will make the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] equal to the identity matrix are [tex]\( x = 0 \)[/tex] and [tex]\( y = -2 \)[/tex].
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